This file contains the mails sent to the GAP forum in Januar-March 1993.

Name               Email address                           Mails    Lines
Martin Schoenert   martin@bert.math.rwth-aachen.de            28     1376
Frank Celler       fceller@bert.math.rwth-aachen.de           17     1294
Joachim Neubueser  neubuese@samson.math.rwth-aachen.de        17      578
Steve Linton       sl25@cus.cam.ac.uk                         11      199
David Sibley       sibley@math.psu.edu                        10      169
Harald Boegeholz   hwb@machnix.mathematik.uni-stuttgart.de     9      295
Jacob Hirbawi      jcbhrb@cerf.net                             7      288
Ralf Dentzer       dentzer@polyhymnia.iwr.uni-heidelberg.de    7      204
Alexander Hulpke   ahulpke@bert.math.rwth-aachen.de            6      251
Thomas Breuer      sam@ernie.math.rwth-aachen.de               5      521
Jean Michel        michel@dmi.ens.fr                           5      249
Ansgar Kaup        kaup@ccucvx.unican.es                       5       58
Arnaldo Mandel     am@ime.usp.br                               4      126
John R. Neil       neil@dehn.mth.pdx.edu                       4       92
Werner Nickel      werner@pell.anu.edu.au                      3       95
Mark Short         short@jordan.csu.murdoch.edu.au             3       77
Leonard Soicher    L.H.Soicher@qmw.ac.uk                       3       39
N. S. Mendelsohn   mendel@ccu.umanitoba.ca                     3       24
Dana-Picard Noah   dana@bimacs.cs.biu.ac.il                    3       22
Dr D. L. Johnson   dlj@maths.nott.ac.uk                        3       15
Eamonn O'Brien     obrien@pell.anu.edu.au                      2       83
Bruce Kaskel       kaskel@math.berkeley.edu                    2       77
Andrea Caranti     caranti@volterra.cineca.it                  2       72
Peter Mueller      muellpe@mi.uni-erlangen.de                  2       47
Lee Schumacher     schumach@math.wisc.edu                      2       36
Derek Holt         dfh@maths.warwick.ac.uk                     2       35
Ulderico Dardano   DARDANO@matna.na.infn.it                    2       33
Borcic Boris       borbor@divsun.unige.ch                      2       24
Volkmar Felsch     felsch@samson.math.rwth-aachen.de           1      137
Ronald Biggers     rbiggers@kscsuna1.kennesaw.edu              1       48
Jane Bamblett      bamblett@maths.ox.ac.uk                     1       38
Frederick Ford     ffor@gauss.math.rochester.edu               1       34
Werenfried Spit    SPIT@vm.ci.uv.es                            1       33
Peter Webb         webb@s5.math.umn.edu                        1       31
Oliver Bonten      oli@ernie.math.rwth-aachen.de               1       27
Martin Wursthorn   pluto@mibtt1.mathematik.uni-stuttgart.de    1       23
Edward Spitznagel  C31801ST@wuvmd.wustl.edu                    1       21
Michael Smith      smith@pell.anu.edu.au                       1       20
Ted Hurley         MATHURLEY@bodkin.ucg.ie                     1       18
Keith Dennis       dennis@math.cornell.edu                     1       18
Lee Hawkins        leh@aberystwyth.ac.uk                       1       15
Meinolf Geck       geck@dmi.ens.fr                             1       14
Anton Greil        greil@guug.de                               1       14
Donald White       white@mcs.kent.edu                          1       13
Dawn Endico        dawn@math.wayne.edu                         1       10
Geoff Smith        gcs@maths.bath.ac.uk                        1        9
Francois Digne     digne@dmi.ens.fr                            1        5
Karl Brodowsky     bro@clio.iwr.uni-heidelberg.de              1        3

This  file is in Berkeley mail drop format, which means you can read this
file with 'mail -f <name-of-the-file>'  or 'mailx -f <name-of-the-file>'.
It is also possible however to read this file with any text editor.



From neubuese@samson.math.rwth-aachen.de Mon Jan  4 13:27:50 1993
From:       neubuese@samson.math.rwth-aachen.de "Joachim Neubueser"
Date:       Mon, 4 Jan 93 13:27:50 +0100
Subject:    Unsolved Problems in Group Theory (Kourovka Notebook) (fwd)

In  a  recent  letter  to  the  GAP-forum,  I  mentioned  the Kourovka
Notebook.  I got the appended detailed  information  about it from Dr.
Khukhro who is presenly in  Freiburg, Germany, which hereby  I want to
make known to all members of the forum.

Happy New Year       Joachim Neubueser

========================================================================
Forwarded message:
> From khukhro@sun8.ruf.uni-freiburg.de Thu Dec 31 14:13:47 1992
> Subject: Unsolved Problems in Group Theory (Kourovka Notebook) (fwd)
> To: neubuese@samson.

> %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
>                                                       Now in English!
>                   UNSOLVED PROBLEMS IN GROUP THEORY
>                         THE KOUROVKA NOTEBOOK
> -----------------------------------------------------------------------
>                The 12-th revised and augmented edition
>                  V.D.Mazurov and E.I.Khukhro, Editors
>              Institute of Mathematics, Novosibirsk, 1992
>                154 p. in A5 format, softcover, 20,- DM
> -----------------------------------------------------------------------
>    This collection of unsolved problems in Group Theory and close areas
> is published regularly, every 2-3 years, starting from 1965. Each new 
> edition is supplemented with new problems and brief comments on the
> solved problems from the previous editions.
>    The problems are proposed by the experts in Group Theory and close
> areas, their level varies from a Ph.D. thesis to long-standing
> well-known problems.
> Among the authors there are many prominent Russian mathematicians, such 
> as S.I.Adian, S.N.Chernikov, Yu.L.Ershov, Yu.M.Gorchakov, R.I.Grigorchuk, 
> M.I.Kargapolov, A.I.Kostrikin, A.I.Mal'cev, Yu.I.Merzliakov,
> A.Yu.Ol'shanskii, V.P.Platonov, B.I.Plotkin, V.N.Remeslennikov,
> L.A.Shemetkov, A.L.Shmel'kin,  V.P.Shunkov, A.E.Zalesskii and others.
> Having acquired internation popularity, the Kourovka Notebook contains
> also problems of R.Baer, G.Baumslag, R.Carter, J.Dixon, R.Griess,
> K.Gruenberg, B.Hartley, H.Heineken, G.Glauberman, O.Kegel,
> B.Neumann, P.Neumann, R.Phillips, C.Praeger, D.Robinson, K.Roggenkamp, 
> J.Roseblade, J.Thompson, G.Wall, B.Wehrfritz, J.Wiegold, H.Wielandt,
> J.Wilson, H.Zieschang and others.
>    Some of the problems are quickly solved due to the fact that they are
> read  by specialists from other fields of Group Theory, others wait for
> substantial breakthroughs for many years. (Of course, it is often easier
> to pose two problems than to solve one.) It is interesting to follow the
> progress in the development of Group Theory, as more and more of these
> problems become solved  (for example, now about 3/4 of the problems from
> the 1-st edition have been solved!).
>    The present book is the 12-th revised and augmented edition (1992), it
> is  now published in two versions, in Russian and in English. The English 
> version is now available for 20,- DM per copy. (This is to cover expenses
> and to improve further editions; since there are now difficulties for
> Russian mathematicians, this kind of support is vital for the further
> existence of the Kourovka Notebook.)
>    Orders (a check, 20,- DM per copy, may be enclosed)
> should be sent to
> 
>     E.I.Khukhro   (e-mail: khukhro@sun1.ruf.uni-freiburg.de 
>                    fax: (49-761)-203-2354)
>         Mathematics Institute of the Freiburg University
>         Albertstrasse, 23 b, Freiburg i.Br., D-7800,  FRG
>   
>                                                    E.I.Khukhro, V.D.Mazurov
> %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   



From dentzer@polyhymnia.iwr.uni-heidelberg.de Mon Jan  4 15:25:32 1993
From:       dentzer@polyhymnia.iwr.uni-heidelberg.de "Ralf Dentzer"
Date:       Mon, 4 Jan 93 15:25:32 +0100
Subject:    Re: GAPstones on Suns

We have some higher GAPstone figures on our Suns, I think this is due to the
gcc compiler we used to make gap, target "sun-sunos-gcc".
With the GAP option -m 2m we get in the first run of combinat.tst:


Model					GAPstones

SLC 	(20 MHz)			15272
1+  	(25 MHz)			17758
4/280 	(16 MHz)			15145
2	(40 MHz)			27888
10-20	(33 MHz)			46876


Ralf Dentzer

dentzer@kalliope.iwr.uni-heidelberg.de



From sl25@cus.cam.ac.uk Mon Jan  4 15:58:37 1993
From:       sl25@cus.cam.ac.uk "Steve Linton"
Date:       Mon, 4 Jan 93 15:58:37 +0100
Subject:    Re: GAPstones on Suns 

Using the standard executable I can add
Sun IPX 	23823
Sum 4/690       23635 (Cypress CY605 processor)

Tonight I'll run it on my PC (386SX/16) and try for a new record
lowest.

	Steve



From sibley@math.psu.edu Tue Jan  5 09:14:31 1993
From:       sibley@math.psu.edu "David Sibley"
Date:       Tue, 5 Jan 93 09:14:31 +0100
Subject:    Character Tables

I have been computing some character tables using CharTablePGroup.  Is
the order of the conjugacy classes in <G>.conjugacyClasses the same as
the order of columns in the table?  The manual implies this, but does
not say it directly.  If not, how do I know which class in the group
goes with which column in the table?

Is there some way to get the (very nice) output of DisplayCharTable
directed to a file?  I've been using cut & paste to save them, but there
must be a better way.  (I realize there is a problem with what width to
use.)

What does the message

  #I TransformingPermutations: no bijection of row families

mean?  (This might not be exactly right.  It's from memory.)  I don't
see it very often, though I've been using TransformingPermutations quite
a bit, so I assume it's something unusual.



From ahulpke@bert.math.rwth-aachen.de Tue Jan  5 10:16:36 1993
From:       ahulpke@bert.math.rwth-aachen.de "Alexander Hulpke"
Date:       Tue, 5 Jan 93 10:16:36 +0100
Subject:    Re: Character Tables

In his letter David Sibley asked:
> Is the order of the conjugacy classes in <G>.conjugacyClasses the same as
> the order of columns in the table?  The manual implies this, but does
> not say it directly.

The order *is* the same, though the manual is somehow evasive in this point.
All routines computing  character tables of groups (i.e. CharTablePGroup and
also the forthcoming Dixon-Algorithm) will use the order of the classes of the
group. The only requirement is, that the class of the identity *must* be in
the first position. If the classes are not yet given, the routines compute
them, and use them in the computed order.
However, be careful with SortClassesCharTable: In GAP 3.1 this will *not*
rearrange the conjugacy classes of the group, given in 'Table.group'. This
will be fixed in GAP 3.2.

> Is there some way to get the (very nice) output of DisplayCharTable
> directed to a file?

I would suggest the following routine, which will do the required task. It
is used by
  
  PrintCharTable(file,table);

or --- alternatively --- by

  PrintCharTable(file,table,width);

where width is the length of the rows of the final output device, e.g. the
printer:

    PrintCharTable := function(arg)
    local file,table,width,screensize;
      file:=arg[1];
      table:=arg[2];
      if Length(arg)>2 then
	width:=arg[3];
      else 
	# this is the standard print width 
	width:=80;
      fi;
      # get screen size
      screensize:=SizeScreen();
      # resize screen
      SizeScreen([width,screensize[2]]);
      PrintTo(file,DisplayCharTable(table));
      # resize screen to standard size
      SizeScreen(screensize);
    end;

> What does the message
> 
>   #I TransformingPermutations: no bijection of row families
> 
> mean?  (This might not be exactly right.  It's from memory.)  I don't
> see it very often, though I've been using TransformingPermutations quite
> a bit, so I assume it's something unusual.
> 
TransformingPremutations first tries to map the rows --- regarded as sets of
their elements --- of the first matrix onto the rows of the second matrix.
The existence of such a mapping is necessary for the existence of
transforming permutations.
The same observation is valid for the columns, thus the same test is done
therefore.
If one of these two tests fail --- which can be checked quite easily ---
the two matrices cannot be equivalent. This will eliminate the need to start
the (expensive) routine, trying to map the matrices onto each other.
The meaning of the printed message is just a 'false', but it is a *fast*
'false'.

I hope this helps,

		  Alexander



From martin@bert.math.rwth-aachen.de Wed Jan  6 12:00:57 1993
From:       martin@bert.math.rwth-aachen.de "Martin Schoenert"
Date:       Wed, 6 Jan 93 12:00:57 +0100
Subject:    Re: GAP on OS/2 2.0

Steve Linton, who did the port of GAP to MS-DOS, wrote to me about the
subject of GAP on OS/2 2.0.

    No it won't.  The DOS extender that lets it run in a flat 32-bit
    memory model is not compatible with OS/2 at all.

    However, an OS/2 version should not be difficult to do, as OS/2
    provides a 32-bit memory model and a C compiler for it. If you can
    find someone with an OS/2 machine (at Essen maybe) it should be no
    harder than a port to another flavour of UNIX. Indeed I vaguely
    recall the OS/2 stuff claiming to be POSIX compatible, in which case
    no work at all should be needed.

        Steve

So, if there is anybody reading this forum who has OS/2 2.0, a C compiler
for it, and who is willing to try the port, contact me.  I can provide
some hints what needs to be done.

Martin

-- .- .-. - .. -.  .-.. --- ...- . ...  .- -. -. .. -.- .-
Martin Sch"onert,   Martin.Schoenert@Math.RWTH-Aachen.DE,  +49 241 804551
Lehrstuhl D f"ur Mathematik, Templergraben 64, RWTH, D 51 Aachen, Germany



From oli@ernie.math.rwth-aachen.de Wed Jan  6 13:03:56 1993
From:       oli@ernie.math.rwth-aachen.de "Oliver Bonten"
Date:       Wed, 6 Jan 93 13:03:56 +0100
Subject:    Re: re commutators

In reply to a question by K. Dennis, J. Neubueser wrote:
> for all sporadic groups. Can one perhaps use character theory also for
> verifying C_2 or C_3? I have no idea,  so here is another question. If
> so, we  have plenty of charactertables  and tools for handling them in
> GAP.

I don't think it is that easy to use character theory to verify C_2 or
higher. Character theory gives results "up to conjugacy in G", which is
sufficient for C_1, but not for C_2. A character theoretic verification
of C_1 essentialy verifies that for given x_1 there is a conjugacy class
(z) in G such that x_1 is in (z^-1)(z). Clearly , C_1 follows from this.
For given x_1, x_2, there is hope to find a conjugacy class (z) such that
x_1 and x_2 both are in (z^-1)(z), but this doesn't prove C_2, because you
don't know if it is "the same z" you get this way for writing x_1 and x_2
as commutators.
 
Nevertheless, GAP has been a very useful experimental tool for finding the
conjugacy class (z) in my thesis, and for a verification of C_2 or higher,
character theory will show you which conjugacy classes may contain the
wanted element z and which can't.

Oliver Bonten

-- 
Heute hack ich, morgen crack ich, uebermorgen hol ich mir dem SysOp sein Login.
Ach wie gut dass niemand weiss, dass ich oli@math.rwth-aachen.de heiss.



From sl25@cus.cam.ac.uk Wed Jan  6 15:04:14 1993
From:       sl25@cus.cam.ac.uk "Steve Linton"
Date:       Wed, 6 Jan 93 15:04:14 +0100
Subject:    Lowest GAPStone rating?

My PC (386SX/16) returns 1607 GAPStones, using GAPEXE.386 and my
stripped (comments and extra spaces removed) libraries. Has anyone
scored less?



From sam@ernie.math.rwth-aachen.de Thu Jan  7 16:17:06 1993
From:       sam@ernie.math.rwth-aachen.de "Thomas Breuer"
Date:       Thu, 7 Jan 93 16:17:06 +0100
Subject:    

In his answer to a question of David Sibley concerning some messages
printed by 'TransformingPermutations' Alexander Hulpke did not tell
the whole truth.

Two rows are regarded to be equivalent if 'Sort' makes them equal.
The equivalence classes of rows of a matrix are called its row families.

Clearly if two matrices are permutation equivalent then every row
family of the first matrix must consist of rows that are equivalent to
the rows in a family of the other matrix, and the cardinalities of these
families must be equal.  The same of course holds for the columns of
the matrices, but moreover it is also true for the restriction to every
row family of the first matrix and its corresponding family in the second
one; this might yield a finer distribution to column families, and the
program in fact computes this one.

These distributions of rows and columns are not just computed to test
some necessary conditions of permutation equivalence, in order to avoid
starting the algorithm whenever these tests fail.  They are an essential
part of the algorithm, namely they provide the translation of the
problem to find transforming permutations into a problem completely
formulated in terms of permutations.  This problem is then solved by one
of the backtrack search algorithms for permutation groups (very similar
to the algorithm for the computation of a centralizer in a given
permutation group).

Thomas Breuer



From bamblett@maths.ox.ac.uk Wed Jan 13 13:15:29 1993
From:       bamblett@maths.ox.ac.uk "Jane Bamblett"
Date:       Wed, 13 Jan 93 13:15:29 +0100
Subject:    OperationHomomorphism

While testing an implementation of an algorithm for finding the p-core of a permutation group, I came acrss the following GAP peculiarity.
Suppose we type the following GAP commands.

     gap> G := Group((1,2,3,4), (5,6,7,8));;
     gap> H1 := Operation(G, [1,2,3,4]);
     Group((1,2,3,4))
     gap> f1 := OperationHomomorphism(G, H1);;

Then, as expected, we get

     gap> Kernel(f1);
     Subgroup(Group((1,2,3,4), (5,6,7,8)), [(5,6,7,8)])

and everything looks fine.

Now suppose that we wish to be a bit twisted and let the above group G act on the set [1..10] (so that we have two fixed points) and that we define H2 and f2 via the GAP commands

     gap> H2 := Operation(G, [1,2,3,4,9,10]);
     Group((1,2,3,4))
     gap> f2 := OperationHomomorphism(G, H2);;

Then we get:

     gap> PreImages(f2, TrivialSubgroup(H2));
     Subgroup(Group((1,2,3,4), (5,6,7,8)), [(5,6,7,8), (1,2,3,4)])
     gap> Kernel(f2) = G;
     true
     gap> Image(f2, (1,2,3,4));
     (1,2,3,4)
     gap> (1,2,3,4) in Kernel(f2);
     true

- a bit peculiar?

Perhaps wishing to compute with G having fixed points as above is being perverse but perhaps it would be good if GAP could deal with perverse situations like this, or at least notify the user of the possibility of strange results. Any comments?

Jane Bamblett



From martin@bert.math.rwth-aachen.de Wed Jan 13 18:07:27 1993
From:       martin@bert.math.rwth-aachen.de "Martin Schoenert"
Date:       Wed, 13 Jan 93 18:07:27 +0100
Subject:    Re: OperationHomomorphism

In her e-mail message of 13-Jan-93 Jane Bamblett writes
    Suppose we type the following GAP commands.

        gap> G := Group((1,2,3,4), (5,6,7,8));;
        gap> H2 := Operation(G, [1,2,3,4,9,10]);
        Group( (1,2,3,4) )
        gap> f2 := OperationHomomorphism(G, H2);;
        gap> Kernel(f2) = G;
        true

    - a bit peculiar?

Yes indeed.  GAP does the following to compute the kernel of 'f2'.  It
sees that 'G' has three orbits on '[1,2,3,4,9,10]', namely '[1,2,3,4]',
'[9]', and '[10]'.  Thus it computes the stabilizers of 'G' on those
orbits (e.g., 'Stabilizer( G, [1,2,3,4], OnTuples )') and the kernel is
the intersection of those stabilizers.  Now the stabilizers are
'Subgroup( G, [ (5,6,7,8) ] )', 'G', and 'G'.  Unfortunately there is a
bug in 'PermGroupOps.Intersection' for the case that one of the operands
('H') is the parent group of the other operand ('G').  In this case it
simply wants to return 'G', but there is a typo, so it returns 'H'
instead.

This problem is already fixed in GAP 3.2.  If you want to fix it in GAP
3.1, change the line 690 in file '<gap-dir>/lib/permbckt.g' from

        K := ShallowCopy( H );

to read

        K := ShallowCopy( G );

Jane Bamblett continues:

    Perhaps wishing to compute with G having fixed points as above is being
    perverse but perhaps it would be good if GAP could deal with perverse
    situations like this, or at least notify the user of the possibility of
    strange results. Any comments?

No, what you tried is not perverse at all.  In GAP an operation must
satisfy the following conditions.  The domain of operation <D> must be a
list without duplicates (it need not be a set, i.e., it need not be
sorted), the image of every point <d> in <D> under each element <g> of
the group <G> must again lie in <D>, the trivial element of <G> must
operate trivially on <D>, and for all elements <g> and <h> of <G>
'(<d>^<g>)^<h> = <d>^(<g>*<h>)'.  The function that computes the image of
a point <d> in <D> under an element <g> in <G> must either be given as
optional third argument, or the standard operation '<d>^<g>' is used
(which can alternatively be specified by 'OnPoints').  Whenever those
conditions hold you can use 'Operation' and 'OperationsHomomorphism' and
the other functions described in chapter 8, no matter how may orbits or
fixpoints <G> has on <D>.  I think the code is usually optimized for the
case that <G> has only one orbit and no fixpoints though.  From a few
experiments it seems that <D> may even be the empty list, but I wouldn't
be willing to bet that this is always allowed.

Martin.

-- .- .-. - .. -.  .-.. --- ...- . ...  .- -. -. .. -.- .-
Martin Sch"onert,   Martin.Schoenert@Math.RWTH-Aachen.DE,  +49 241 804551
Lehrstuhl D f"ur Mathematik, Templergraben 64, RWTH, D 51 Aachen, Germany



From neil@dehn.mth.pdx.edu Thu Jan 14 21:38:42 1993
From:       neil@dehn.mth.pdx.edu "John R. Neil"
Date:       Thu, 14 Jan 93 21:38:42 +0100
Subject:    Using GAP in a PC Lab Environment

I was wondering if anyone has had any success in getting GAP to operate in
a lab environment.  I have a lab filled with 30 PC workstations and 30 Mac
workstations all running of a single Novell file server.  The PC's are all
386 machines with 4MB of memory.  When I try to run GAP, everything runs fine
until I try to define a group.  Then it just sits there for long periods of
time not doing anything.  It also is not interruptable.  Is this just a
feature of running the MSDOS version of GAP in a network environment?  If
anyone has had success using GAP off of a Novell network, I'd appreciate
hearing what you've done.

--John Neil




=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
John Neil, Graduate Teaching Assistant           e-mail:  neil@math.mth.pdx.edu
Mathematics Department                         NeXTMail:  neil@dehn.mth.pdx.edu
Portland State University
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=



From dlj@maths.nott.ac.uk Fri Jan 15 12:22:56 1993
From:       dlj@maths.nott.ac.uk "Dr D. L. Johnson"
Date:       Fri, 15 Jan 93 12:22:56 +0100
Subject:    Re:  a question to HNN extensions

Dear Gap-forum,
especially Toni. Yes, this is correct. The stable letter in an HNN-extension
appears with exponent-sum zero in every relator and so has infinite order
even in the abelianised group.

Dave Johnson



From dlj@maths.nott.ac.uk Fri Jan 15 12:30:52 1993
From:       dlj@maths.nott.ac.uk "Dr D. L. Johnson"
Date:       Fri, 15 Jan 93 12:30:52 +0100
Subject:    Re:  group theory discussion

Dear Gap-forum,
There is an electronic forum for group-theoretical questions. It is called 
the "pub" and is run by Geoff Smith at Bath, UK.

Dave Johnson



From muellpe@mi.uni-erlangen.de Fri Jan 15 17:56:44 1993
From:       muellpe@mi.uni-erlangen.de "Peter Mueller"
Date:       Fri, 15 Jan 93 17:56:44 +0100
Subject:    Primitive Groups

The list PGTable in the primitive groups library contains some useful
data about the primitive groups of degree \le 50.

In practice one often likes to know which class of the O'Nan Scott
classification a primitive group belongs to. Wouldn't it be sensible
to add this information as a further component? Additionally,
generators of one of the (at most two) minimal normal subgroups would
be fine, as GAP doesn't seem (or did I miss something?) to offer a
convenient way to compute them.

Peter

================================================================
Peter M\"uller
Math. Institut
Univ. Erlangen-N\"urnberg
Bismarckstr. 1 1/2
8520 Erlangen



From neubuese@samson.math.rwth-aachen.de Fri Jan 15 18:03:14 1993
From:       neubuese@samson.math.rwth-aachen.de "Joachim Neubueser"
Date:       Fri, 15 Jan 93 18:03:14 +0100
Subject:    Re:  group theory discussion

Dear David,
thanks for your two remarks in  the  gap-forum. In fact Geoff had also
himself mentioned pub in a letter to  the gap-forum and I for one have
asked to be  enrolled to it, but so far did not get a message from it.
If you get any, please send me a copy.

Kind regards   Joachim



From gcs@maths.bath.ac.uk Fri Jan 15 19:12:15 1993
From:       gcs@maths.bath.ac.uk "Geoff Smith"
Date:       Fri, 15 Jan 93 19:12:15 +0100
Subject:    Re: group theory discussion

PLease do not inundate Joachim with copies of 
group-pub-forum@maths.bath.ac.uk discussion.

His enrollment request disappeared down a hole somewhere. It never reached
me. He is now enrolled and has been sent recent group-pub-forum
correspondence.

Geoff Smith



From greil@guug.de Tue Jan 19 11:43:48 1993
From:       greil@guug.de "Anton Greil"
Date:       Tue, 19 Jan 93 11:43:48 +0100
Subject:    Re: group theory discussion

> 
> PLease do not inundate Joachim with copies of 
> group-pub-forum@maths.bath.ac.uk discussion.
> 
> Geoff Smith
> 

Has me question about a conjugacy problem last week to the group-pub-forum
produced such copies? Then there may be a bug somewhere in the configuration
of the two mailing-lists.

Regards, Anton Greil
greil@guug.de



From neubuese@samson.math.rwth-aachen.de Tue Jan 19 12:31:22 1993
From:       neubuese@samson.math.rwth-aachen.de "Joachim Neubueser"
Date:       Tue, 19 Jan 93 12:31:22 +0100
Subject:    Re: group theory discussion

No worry, there seems to be everything under control in the gap-forum,
I certainly was not inundated by mail.
Thanks for everybody's concern 
Joachim Neubueser



From martin@bert.math.rwth-aachen.de Thu Jan 21 13:58:10 1993
From:       martin@bert.math.rwth-aachen.de "Martin Schoenert"
Date:       Thu, 21 Jan 93 13:58:10 +0100
Subject:    Re: Primitive Groups

In his article of 1993/01/15 Peter M"uller writes:
    The list PGTable in the primitive groups library contains some useful
    data about the primitive groups of degree \le 50.

    In practice one often likes to know which class of the O'Nan Scott
    classification a primitive group belongs to. Wouldn't it be sensible
    to add this information as a further component? Additionally,
    generators of one of the (at most two) minimal normal subgroups would
    be fine, as GAP doesn't seem (or did I miss something?) to offer a
    convenient way to compute them.

Several comments.

1)  Sims says in his article "Computational methods in the study of
    permutation groups" in "Computational Problems in Abstract Algebra"
    (Proceedings Oxford 67, John Leech ed.), which contains the primitive
    groups of degree <= 20:

        Any primitive group G of degree n < 60 has a unique minimal
        normal subgroup N.

    So how comes you talk about "the (at most two) minimal normal
    subgroups"?  Or am I misunderstanding something here?

2)  If the primitive group G of degree p^k has an elementary abelian
    regular normal subgroup, this is can be computed as

        Core( G, SylowSubgroup( G, p ) )

    Peter Neumann in his article "Some algorithms for computing with
    finite permutation groups" in "Proceedings of Groups - St. Andrews
    1984" (E.F. Robertson & C.M. Campbell ed.) gives a better algorithm,
    which avoids the computation of the Sylow subgroup.  But for the
    groups in GAP's primitive group library this simple approach seems to
    be ok.

3)  If the primitive group G of degree p^k < 5^5 is not of the affine
    type the socle (i.e., because of 1) the minimal normal subgroup)
    is the last term in the derived series, and can be computed as

        d := DerivedSeries( g );
        d[Length(d)];

4)  The library file 'grp/primitiv.grp' identifies the minimal normal
    subgroup, if it is again primitiv.  For example for

        g := PrimitiveGroup( 28, 11 );

    the corresponding line in 'grp/primitiv.grp' reads

        [28,     29484,  2, 01000, 2805, 2707,  "PZL(2,27)", 148,151,156,171],

    it tells us (in encoded form in the 5. column) that the minimal
    normal subgroup is 'PrimitiveGroup( 28, 5 )', and thus we get is as

        AsSubgroup( g, PrimitiveGroup( 28, 5 ) );

    If the fifth entry reads 'EABL' the minimal normal subgroup is
    elementary abelian, and thus we can apply 2).  If the fifth entry
    is empty the minimal normal subgroup is not elementary abelian and
    not primitive, and thus we should apply 3).

Hope this helps, Martin.

-- .- .-. - .. -.  .-.. --- ...- . ...  .- -. -. .. -.- .-
Martin Sch"onert,   Martin.Schoenert@Math.RWTH-Aachen.DE,  +49 241 804551
Lehrstuhl D f"ur Mathematik, Templergraben 64, RWTH, D 51 Aachen, Germany



From dfh@maths.warwick.ac.uk Thu Jan 21 23:48:34 1993
From:       dfh@maths.warwick.ac.uk "Derek Holt"
Date:       Thu, 21 Jan 93 23:48:34 +0100
Subject:    Re: Primitive Groups

With reference to the recent discusssion on primitive groups, the
smallest such group (in terms of both order and degree) which has two
distinct minimal normal subgroups is in fact  A5 x A5  in its
permutation representation of degree 60 on a diagonal subgroup.

Derek Holt.



From C31801ST@wuvmd.wustl.edu Tue Jan 26 05:10:39 1993
From:       C31801ST@wuvmd.wustl.edu "Edward Spitznagel"
Date:       Tue, 26 Jan 93 05:10:39 +0100
Subject:    FactorsInt and 6th Fermat Number

I tried factoring the 6th Fermat number using the code below and
got F6 returned to me, suggesting that F6 is prime.  Since one
factor of F6 is 274177, this seems to contradict Section 10.20
of the documentation, which states that "FactorsInt is
guaranteed to find all factors less than 10^6."

           gap> FactorsInt( 2^(2^6) + 1);
           [ 18446744073709551617 ]
           gap> (2^(2^6) + 1) / 274177;
           67280421310721
           gap> quit;

I doubt that it makes any difference, but my hardware is a Turbo
Color NeXTstation running NeXTSTEP 2.1.  My version of GAP is 3.1,
obtained as gapexe.next from SAMSON.

Edward Spitznagel
Department of Mathematics
Washington University
c31801st@wuvmd.wustl.edu



From kaup@ccucvx.unican.es Tue Jan 26 14:00:12 1993
From:       kaup@ccucvx.unican.es "Ansgar Kaup"
Date:       Tue, 26 Jan 93 14:00:12 +0100
Subject:    FactorsInt and 6th Fermat Number

I tried factoring the 6th Fermat number in order to check the result that 
Edward Spitznagel got with his NeXT and FactorsInt found the correct factors.
I am using a SUN SPARC station 10 and got the executable from SAMSON aswell.
So the bug must have something to do with Edwards Hardware or is contained in
the executable obtained from SAMSON.

Ansgar Kaup
departamento de matematicas
Universidad de Cantabria, Santander
kaup@ccucvx.unican.es



From sl25@cus.cam.ac.uk Tue Jan 26 14:57:50 1993
From:       sl25@cus.cam.ac.uk "Steve Linton"
Date:       Tue, 26 Jan 93 14:57:50 +0100
Subject:    Re: FactorsInt and 6th Fermat Number 

I think this problem has come up already. There was a bug in the library,
for which Martin posted a patch. The patch may well be in the 3.1 patchlevel
3 library, so I suggest you get that if you are not using it already. Otherwise
mail me directly and I'll fish out the patch from my mail logs.

	Steve Linton



From martin@bert.math.rwth-aachen.de Tue Jan 26 17:40:11 1993
From:       martin@bert.math.rwth-aachen.de "Martin Schoenert"
Date:       Tue, 26 Jan 93 17:40:11 +0100
Subject:    Re: FactorsInt and 6th Fermat Number

In his e-mail message of 1993/01/26 Edward Spitznagel wrote:
    I tried factoring the 6th Fermat number using the code below and
    got F6 returned to me, suggesting that F6 is prime.  Since one
    factor of F6 is 274177, this seems to contradict Section 10.20
    of the documentation, which states that "FactorsInt is
    guaranteed to find all factors less than 10^6."

           gap> FactorsInt( 2^(2^6) + 1);
           [ 18446744073709551617 ]
           gap> (2^(2^6) + 1) / 274177;
           67280421310721
           gap> quit;

This is indeed a bug in the function 'IsPrime'.  Since 'IsPrime(2^64+1)'
returns 'true', 'Factors' doesn't even attempt to factor it.  A fix for
that bug is in the upgrade from GAP 3.1 patchlevel 2 to GAP 3.1
patchlevel 3, which is available as 'upg3r1p3.dif.Z' from
'samson.math.rwth-aachen.de'.

More about the problem (i.e., the reason why 'IsPrime' failed), can be
found in one of my e-mails to the GAP Forum, which is in the file
'forum92d.txt' (GAP Forum mails of 4th quarter 92), on
'samson.math.rwth-aachen.de' in the directory 'tmp' (not in 'pub/gap' as
the rest of GAP).

In his e-mail message of 1993/01/26 Ansgar Kaup answered

    I tried factoring the 6th Fermat number in order to check the result that 
    Edward Spitznagel got with his NeXT and FactorsInt found the correct factors.
    I am using a SUN SPARC station 10 and got the executable from SAMSON aswell.
    So the bug must have something to do with Edwards Hardware or is contained in
    the executable obtained from SAMSON.

I think the NeXT people would be quite offended by the idea that their
hardware is to blame ;-).  (Oh, and of course the executables we
distribute never have bugs.  Well, hardly ever ;-)

Martin.

-- .- .-. - .. -.  .-.. --- ...- . ...  .- -. -. .. -.- .-
Martin Sch"onert,   Martin.Schoenert@Math.RWTH-Aachen.DE,  +49 241 804551
Lehrstuhl D f"ur Mathematik, Templergraben 64, RWTH, D 51 Aachen, Germany



From martin@bert.math.rwth-aachen.de Tue Jan 26 18:18:15 1993
From:       martin@bert.math.rwth-aachen.de "Martin Schoenert"
Date:       Tue, 26 Jan 93 18:18:15 +0100
Subject:    GAPstones

Here is a selected list of GAPstones for various machines.
Computer     Model  MHz Memory  OS      Compiler    GAPstones   by

Atari        ST520+  8    4 MB  TOS     gcc 2.1      1168       M. Schoenert

Apollo       DN10000 ?       ?  SR10.3  ?           20625       R. Lewis

DECstation   3100   20   32 MB  Ultrix  gcc 2.2     22590       M. Schoenert
DECstation   5120   25   32 MB  Ultrix  gcc 2.2     24942       M. Schoenert

IBM PC       386SX  16       ?  DOS     djgpp 1.05   1607       S. Linton
IBM PC       386DX  33       ?  DOS     djgpp 1.05   6057       S. Rosenbrock
IBM PC       486DX  33       ?  DOS     djgpp 1.05   9948       L. Soicher
IBM PC       486DX  50   16 MB  386BSD  gcc 2.3.3   32894       M. Schoenert
IBM PC       486DX  50   16 MB  DOS     djgpp 1.09 ~32000       M. Schoenert

IBM RS6000   950   ~40    lots  AIX     cc          41691       R. Dentzer

HP           730    67  128 MB  HP-UX 8 cc          63578       M. Schoenert

NeXTstation         25       ?  Mach    ?           16484       G. Mess

Sparc        4/280  40   32 MB  SunOs   gcc 2.1     15145       R. Dentzer
SparcStation SLC     ?       ?  SunOs   ?           11962       M. Smith
SparcStation SLC     ?   12 MB  SunOS   ?           12477       D. Endico
SparcStation SLC    20   16 MB  SunOs   gcc 2.1     15272       R. Dentzer
SparcStation 1+      ?       ?  SunOs   ?           14203       W. Nickel
SparcStation 1+      ?       ?  SunOs   ?           15032       M. Smith
SparcStation 1+      ?       ?  SunOs   ?           16739       D. Sibley
SparcStation 1+     25   24 MB  SunOs   ?           17758       R. Dentzer
SparcStation ELC     ?       ?  SunOs   ?           19821       M. Smith
SparcStation IPX     ?   64 MB  SunOS   ?           23782       D. Endico
SparcStation 2       ?       ?  SunOS   ?           24013       A. Caranti
SparcStation 2      40   32 MB  SunOS   gcc 2.1     27888       R. Dentzer
SparcStation ?       ?   64 MB  SunOS   ?           39772       A. Kaup
SparcStation 10-20  33   32 MB  SunOS   gcc 2.1     47619       R. Dentzer
SparcStation 10-31   ?       ?  SunOs   ?           44917       M. Smith
SparcStation 10-30   ?       ?  SunOS   ?           48329       W. Nickel

A few remarks:

The Atari ST520+ is the slowest machine.  It is more than a factor of 50
slower than the fastest machine (a HP 730).  Note that quite a bit of
development of GAP was done on this machine (it is the one I have at
home).  Compilation (with optimization) of the entire GAP kernel takes
over 4 hours on this machine.

There is a big difference between L. Soicher's result for a 486DX33 and
our result for 486DX50.  One would expect that the 486DX50 is 50% faster,
but not that it is more that 3 times faster.  The reason is the
following.  In regular intervals (actually whenever a loop body is
executed) GAP checks whether the user wants to interrupt the computation.
Under UNIX this means checking a variable that is set by the signal
handler for '<ctr>-C'.  Unfortunatly this is not possible under DOS/GO32
(or is it? 'setcbrk' might do what we want, but the documentation of
DJGPP only says that there is such a function, not what it does).  Thus
under DOS GAP checks whether the user has pressed a key, and if so,
whether it was '<ctr>-C'.  Now checking whether a key was pressed
requires a system call ('kbhit()'), which means that GO32 (the DOS
extender) has to switch from protected mode (in which GAP runs) to real
mode (which DOS wants).  This is a *very* expensive operation.  So
expensive, that GAP 3.1 spent more than half of the time it took to
execute 'combinat.tst' switching from protected mode to real mode and
back again.  In 3.2 we have changed GAP so that only every 20th check
actually does call 'kbhit()'.  This made GAP more than a factor of 2
faster.  And the time between pressing '<ctr>-C' and GAP responding
is still small enough.

The results for SparcStations vary somewhat.  I think this has to do with
the compiler used.  GNU CC 2.x seems to be best.

The NeXTstation is actually faster than the above result would indicate.
Namely, one has to use GNU CC 2.x, instead of the compiler that comes
with the operating system, which is GNU CC 1.39 (I think).  However,
installation of GNU CC 2.x is not a trivial task (or so I am told, there
seems to be some problem with precompiled header files).

Martin.

-- .- .-. - .. -.  .-.. --- ...- . ...  .- -. -. .. -.- .-
Martin Sch"onert,   Martin.Schoenert@Math.RWTH-Aachen.DE,  +49 241 804551
Lehrstuhl D f"ur Mathematik, Templergraben 64, RWTH, D 51 Aachen, Germany



From kaup@ccucvx.unican.es Wed Jan 27 10:37:43 1993
From:       kaup@ccucvx.unican.es "Ansgar Kaup"
Date:       Wed, 27 Jan 93 10:37:43 +0100
Subject:    Sorry

Sorry for not believing in executables distributed not containing bugs
but I didn't think of not-applicated updates.

Ansgar Kaup



From neil@dehn.mth.pdx.edu Wed Jan 27 17:12:23 1993
From:       neil@dehn.mth.pdx.edu "John R. Neil"
Date:       Wed, 27 Jan 93 17:12:23 +0100
Subject:    Running GAP on MSDOS systems

A couple of weeks ago I posted to this group the problems I was experiencing
running GAP in our lab full of PC's on a Novell network.  The problem turned
out to be completely unrelated to the use of a network.  It turns out that
GAP is unable to run correctly if EMM386 is in the CONFIG.SYS file.  The
GO32 extender is incompatible with this extended memory management tool.  A
warning should perhaps be placed in the documentation warning users of this
conflict.  I have removed this device driver from the PC's in my lab and now
GAP runs perfectly (at, of course, 386-25 speed...).

--John Neil




=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
John Neil, Graduate Teaching Assistant           e-mail:  neil@math.mth.pdx.edu
Mathematics Department                         NeXTMail:  neil@dehn.mth.pdx.edu
Portland State University
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=



From martin@bert.math.rwth-aachen.de Wed Jan 27 17:52:06 1993
From:       martin@bert.math.rwth-aachen.de "Martin Schoenert"
Date:       Wed, 27 Jan 93 17:52:06 +0100
Subject:    Re: Running GAP on MSDOS systems

I can't check right now, but I'm pretty certain that we run GAP with
'EMM386' in 'config.sys'.  Somewhere in the documentation for 'djgpp' it
says that you cannot use the 'noems' switch, whatever that means.  Also I
think that if GAP has problems with the memory manager it would not even
start, while you report that the problems appear once GAP tries to read a
file.  Of course if GAP now works in your lab, you may not want to
experiment further (if it ain't broke ...), on the other hand I wonder
how much memory GAP can use if you run no memory manager.  Maybe only 640
kByte, and when it needs more it starts paging (ouch)?

Martin.

PS. I have this theory about why Bill Gates is so rich.  Secretly every
programmer who makes a living from writing a program that fixes a problem
or a deficiency in MSDOS has to pay Bill Gates 10% of his income.  Well
maybe it's only 1%.

-- .- .-. - .. -.  .-.. --- ...- . ...  .- -. -. .. -.- .-
Martin Sch"onert,   Martin.Schoenert@Math.RWTH-Aachen.DE,  +49 241 804551
Lehrstuhl D f"ur Mathematik, Templergraben 64, RWTH, D 51 Aachen, Germany



From sl25@cus.cam.ac.uk Fri Jan 29 16:05:28 1993
From:       sl25@cus.cam.ac.uk "Steve Linton"
Date:       Fri, 29 Jan 93 16:05:28 +0100
Subject:    Re: Running GAP on MSDOS systems 

-------- 
There are two or three different programs EMM386 around. In
particular the one you get with Windows 3.00 is deficient. You
should replace it with the one that comes with DOS 5.00a. Better
still get the Quarterdeck product QEMM386.

	Steve



From neil@dehn.mth.pdx.edu Fri Jan 29 22:37:44 1993
From:       neil@dehn.mth.pdx.edu "John R. Neil"
Date:       Fri, 29 Jan 93 22:37:44 +0100
Subject:    Re: Running GAP on MSDOS systems 

In message <m0nHxIy-00002hC@bootes.cus.cam.ac.uk> you write:
>-------- 
>
>There are two or three different programs EMM386 around. In
>particular the one you get with Windows 3.00 is deficient. You
>should replace it with the one that comes with DOS 5.00a. Better
>still get the Quarterdeck product QEMM386.
>
>	Steve
>

We were using the one which comes with DOS 5.0.  The main problem with
EMM386 (no matter who's you use) is that their behavior, particularly with
many TSR's running (as  you need to have when attached to a network), is 
extremely unreliable and unpredictable.  Problems will work just fine until
they wish to access EMM and then they crash.  Or sometimes they work reliably
for the first several calls and then crash.  We have found that in many cases
if a program is having any problems that are memory related, get rid of
whatever version of EMM386 you are using and everything will work just fine.
As far as GAP starting to page, I believe that DJGPP has it's own memory
management stuff built in to the 32-bit extender.  Thus, EMM386 is totally
unnecessary anyway.

--John Neil

PS.  386MAX causes GAP to immediately lock up upon execution.




=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
John Neil, Graduate Teaching Assistant           e-mail:  neil@math.mth.pdx.edu
Mathematics Department                         NeXTMail:  neil@dehn.mth.pdx.edu
Portland State University
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=



From dentzer@polyhymnia.iwr.uni-heidelberg.de Mon Feb  1 10:06:51 1993
From:       dentzer@polyhymnia.iwr.uni-heidelberg.de "Ralf Dentzer"
Date:       Mon, 1 Feb 93 10:06:51 +0100
Subject:    Re: GAPstones

Just a little correction. The line
Sparc        4/280  40   32 MB  SunOs   gcc 2.1     15145       R. Dentzer

should read

Sparc        4/280  16   40 MB  SunOs   gcc 2.1     15145       R. Dentzer

i.e. the machine has 40 MB of main memory and a frequency of 16 MHz.

And as I am just at it, do the authors of "combinat.tst" know
what is mainly measured by this benchmark? From a gprof it seems
that most of the time is spent in the GAP Interpreter (>50 %)
and only very little for doing arithmetic and memory management (<5 % each).
Does this reflect time usage of typical GAP routines? What about
permutations, AG groups, finite fields, cyclotomic fields?

Thanks

Ralf Dentzer		dentzer@kalliope.iwr.uni-heidelberg.de



From martin@bert.math.rwth-aachen.de Tue Feb  2 17:22:36 1993
From:       martin@bert.math.rwth-aachen.de "Martin Schoenert"
Date:       Tue, 2 Feb 93 17:22:36 +0100
Subject:    Re: Re: GAPstones

In his e-mail message of 1993/02/01 Ralf Dentzer writes:
    And as I am just at it, do the authors of "combinat.tst" know
    what is mainly measured by this benchmark? From a gprof it seems
    that most of the time is spent in the GAP Interpreter (>50 %)
    and only very little for doing arithmetic and memory management (<5 % each).
    Does this reflect time usage of typical GAP routines? What about
    permutations, AG groups, finite fields, cyclotomic fields?

It is quite difficult to say what a typical use of GAP is.  For example,
if one works with permutation groups of degree over 100, one sees an
entire different picture.  On the other hand, computations with character
tables that involve only very few irrationalities might actually show a
similar distribution (I never tried this though).

And the answer is, yes, we are aware of the specific distribution of
"combinat.tst".  But we have found that the performance reported by
"combinat.tst" is reasonably close to our ``feeling'' of how fast the
machines we have tried it on are.


Actually one can even be a bit more specific.  Here is the output of
'pixie' (the most accurate profiling tool, because it actually counts
cycles, unfortunately only available on machines with the MIPS processor)
for GAP 3.1 running "combinat.tst".  The first column contains the number
of cycles, the second the percentage of the total number of cycles, the
third the number of calls, and the fourth the name of the function.  I
have grouped the functions according to the package they belong to, and
have ommitted functions whose percentage was below 0.1%.

Interpreter (eval 22.08%, function 19.83%, statemen 14.59%)

  33595913    7.65 	   2584301 EvVar (eval.c)
   9776625    2.23 	    196075 Eq (eval.c)
   9435970    2.15 	    190798 Diff (eval.c)
   9324288    2.12 	    181912 Sum (eval.c)
   8133406    1.85 	     71330 FunShallowCopy (eval.c)
   6900221    1.57 	    188588 EvVarAss (eval.c)
   6238983    1.42 	    143309 EvAnd (eval.c)
   5070936    1.15 	    101630 EvOr (eval.c)
   2867693    0.65 	     39239 Prod (eval.c)
   2094182    0.48 	     38040 Ne (eval.c)
   1875496    0.43 	     33166 Le (eval.c)
    435759    0.10 	      6307 Mod (eval.c)

  55270007   12.59 	    250223 EvFunccall (function.c)
  28437062    6.48 	    183380 ChangeEnv (function.c)
   3205685    0.73 	     91591 EvReturn (function.c)

  38293896    8.72 	    266957 EvIf (statemen.c)
  17140123    3.90 	     53408 EvFor (statemen.c)
   8393290    1.91 	     89811 EvStatseq (statemen.c)

Memory Management (gasman 21.51%)

  33964602    7.74 	    308242 NewBag (gasman.c)
  32295304    7.36 	        10 CollectGarb (gasman.c)
  11116554    2.53 	    362625 ExitKernel (gasman.c)
   8523620    1.94 	     57558 Resize (gasman.c)
   4714125    1.07 	    362625 EnterKernel (gasman.c)
   2938674    0.67 	    186597 NrHandles (gasman.c)
    873862    0.20 	         1 InitGasman (gasman.c)

Lists (list 16.99%, vector 0.02%)

  31478498    7.17 	    379801 EvListElm (list.c)
  19866656    4.52 	    162053 EvListAss (list.c)
  13102065    2.98 	     60932 FunAppend (list.c)
   5375002    1.22 	     68101 MakeList (list.c)
   1429617    0.33 	     68077 EvMakeList (list.c)
   1249837    0.28 	     12062 FunAdd (list.c)
    946479    0.22 	      4636 PosList (list.c)

Arithmetic (integer 1.21%, rational 0.17%)

   1469833    0.33 	     24474 QuoInt (integer.c)
   1396376    0.32 	     27243 ProdInt (integer.c)
   1194533    0.27 	     16211 GcdInt (integer.c)
    524400    0.12 	      4930 SumInt (integer.c)

    423835    0.10 	      4678 QuoRat (rational.c)

Reading (scanner 1.38%, read 0.54%, indents 0.2%, system 0.54%, libc 0.87%)

   2086621    0.48 	     19636 GetSymbol (scanner.c)
   1383089    0.31 	      7972 GetIdent (scanner.c)
    924253    0.21 	     24385 PutChr (scanner.c)
    745285    0.17 	      5104 Pr (scanner.c)
    881700    0.20 	      5460 FindIdent (idents.c)
   1133070    0.26 	    226614 SyIsIntr (system.c)
    855227    0.19 	      4319 SyFgets (system.c)
   2982948    0.68 	      4317 fgets (../fgets.c)


A total of 56.5% of the running time are spent in the interpreter.

There most of the time is spent evaluating function calls ('EvFunccall'
and 'ChangeEnv').  This is because most of the functions in the
combinatorics package work in a highly recursive manner, and thus the
number of function calls is unusually high.

(Looking at the number of calls to 'ChangeEnv' (which is called twice by
'EvFunccall' for every non-internal GAP function), one sees that 91690
calls are calls to GAP functions, and the remaining 158533 are calls to
internal functions: 71330 'ShallowCopy', 60932 'Append', 12062 'Add',
5439 'Length', 4004 'Position', 2071 'IsList', 1976 'IsFunc', and 719
others.  Comparing the numbers for 'ChangeEnv' and 'EvReturn' we see that
there were 99 calls to functions which did not return anything, most
likely they are all calls to 'Sort'.  Finally note that both 'EvIf' and
'EvFor' have 'EvStatseq' inlined, i.e., they do not call 'EvStatseq' to
evaluate their bodies.  So the number of calls of 'EvStatseq' reflects
the number of calls to GAP functions that have more than a single
statement in their body.  So comparing the numbers for 'ChangeEnv' and
'EvStatseq' we see that about 1880 calls where to functions with only a
single statement in their body, e.g., functions of the form '<var> ->
<expr>'.)

'EvVar' contributes a large percentage of the total running time, simply
because it is call so often.  Improving the speed of 'EvVar' would
therefor speed up GAP quit a bit.  Unfortunately 'EvVar' is already as
minimal as it can be.

All in all the amount of time spent in the interpreter is much higher
than one would expect from other computations.  Especially the large
number of function calls is not typical.


The memory management uses 21.5% of the total running time.  288655 of
the 308242 created bags ('NewBag') are easily accounted for: 91690 from
'EvFunccall', 71330 from 'ShallowCopy', 68077 from 'MakeList', and 57558
from 'Resize' calls.  Note that the number of calls to 'Resize' is again
unusually high.

(We are thinking about a new memory manager (Ralf has actually provided a
prototype for such a memory manager).  This new memory manager will make
'NewBag' a lot faster (probably a factor of 5 or so).  'NewBag' will also
become a macro, so that it will be more difficult to see its influence in
future profiles.  'CollectGarb' will become faster (because it will use
generations), 'ExitKernel' and 'EnterKernel' will no longer be
neccessary, and the number of calls to 'NewBag' will also be reduced
(because it is called most of the time from 'CollectGarb').  So I think
that the cost for memory management for this test could be reduced to
well below 10% with this new memory manager.)

Other computations will usually spent less time in the memory management,
unless most of the objects allocated are rather small, such as in this
computation.  And of course, if the workspace is too small, so that too
many garbage collections are neccessary, the percentage of time spent in
'CollectGarb' can become arbitrarily high (I remember a Schreier Sims
computation which I once did on my Atari, where a garbage collection was
neccessary each time after allocating 20 permutations).


Now "combinat.tst" mainly works with lists (of integers), thus it is not
surprising that 17% of the total time is spent in the list package.
Especially the number of assignments to list elements is way above the
norm.

Again all in all one would expect other computations to spent less time
in the list package.  But since lists are the main data structure in GAP
almost all computations spent quite a bit of their time in this package.


The amount of time spent in the arithmetic is small.  This is because
arithmetic of immediate integers (i.e., -2^28 <= n < 2^28) is done in
'Eq', 'Sum', 'Diff', etc., without calling other functions.  What you see
in 'SumInt', etc. are only the few computations that involve large
integers (e.g., Binomial( 400, 50 )).

Other computations that involve a lot of large integers, rationals,
cyclotomics, permutations, words, etc., will probably spent most of their
time in the respective arithmetic package.


And of course the time spent reading is very small.  If you only do a
small computation with a group (where the whole library must first be
read in), this would contribute a much higher amount.


Martin.

-- .- .-. - .. -.  .-.. --- ...- . ...  .- -. -. .. -.- .-
Martin Sch"onert,   Martin.Schoenert@Math.RWTH-Aachen.DE,  +49 241 804551
Lehrstuhl D f"ur Mathematik, Templergraben 64, RWTH, D 51 Aachen, Germany



From L.H.Soicher@qmw.ac.uk Thu Feb  4 15:57:23 1993
From:       L.H.Soicher@qmw.ac.uk "Leonard Soicher"
Date:       Thu, 4 Feb 93 15:57:23 +0100
Subject:    new version of GRAPE

       GRAPE  (GRaph Algorithms using PErmutation groups)
A new version of GRAPE is available via anonymous ftp from 
IP number 138.37.80.15.  It is in the pub directory in files
grape.README and grape.tar.Z. Please read grape.README.
Also, GRAPE includes some TeX documentation to get you started.
If you collect this version of GRAPE and set it up (or have problems 
setting it up), please email me to this effect.

Notes: This version of GRAPE is an improved and expanded version of 
that released in August, 1992. All known bugs are fixed: the only
bug which was not minor was in QuotientGraph, but this bug should not 
have caused incorrect results when the quotient map was a covering map 
or when the group associated with the graph to be quotiented was 
trivial.

L.H.Soicher@qmw.ac.uk



From muellpe@mi.uni-erlangen.de Sun Feb  7 18:55:03 1993
From:       muellpe@mi.uni-erlangen.de "Peter Mueller"
Date:       Sun, 7 Feb 93 18:55:03 +0100
Subject:    Normalizer doesn't normalize?

Do I misunderstand something?

gap> Parent(g)=Parent(hp);
true
gap> g=Parent(g);
true
gap> np:=Normalizer(g,hp);;
gap> IsNormal(np,hp);
false

================================================================

gap> g;
Group( ( 1, 7,12,16,19,21, 6)( 2, 8,13,17,20, 5,11)( 3, 9,14,18, 4,10,15), 
( 1, 2)( 3, 6)( 8,15)( 9,13)(10,14)(11,12)(16,20)(17,21), ( 1,19)( 2,21)
( 3,15)( 4,20)( 5,14)( 6,13)( 7,17)(11,16)(12,18) )
gap> hp;
Subgroup( Group( ( 1, 7,12,16,19,21, 6)( 2, 8,13,17,20, 5,11)( 3, 9,14,18, 4,
 10,15), ( 1, 2)( 3, 6)( 8,15)( 9,13)(10,14)(11,12)(16,20)(17,21), ( 1,19)
( 2,21)( 3,15)( 4,20)( 5,14)( 6,13)( 7,17)(11,16)(12,18) ), 
[ ( 1,15,16, 5,11,13,17)( 2,18, 4,21, 9,14, 8)( 3, 6,20,19,10, 7,12), (), 
  ( 2, 9)( 3,20)( 5,11)( 7,10)( 8,14)(12,19)(13,16)(15,17)(18,21), 
  ( 1, 3,21,11,12,14)( 2, 5, 6,18,15, 7)( 4,17,20, 8,13,10)( 9,16,19) ] )
gap> quit;


Peter



From neubuese@samson.math.rwth-aachen.de Mon Feb  8 13:53:03 1993
From:       neubuese@samson.math.rwth-aachen.de "Joachim Neubueser"
Date:       Mon, 8 Feb 93 13:53:03 +0100
Subject:    Re: Normalizer doesn't normalize?

Peter Mueller writes:

> 
> Do I misunderstand something?
> 
> gap> Parent(g)=Parent(hp);
> true
> gap> g=Parent(g);
> true
> gap> np:=Normalizer(g,hp);;
> gap> IsNormal(np,hp);
> false
> 
> ================================================================
> 
> gap> g;
> Group( ( 1, 7,12,16,19,21, 6)( 2, 8,13,17,20, 5,11)( 3, 9,14,18, 4,10,15), 
> ( 1, 2)( 3, 6)( 8,15)( 9,13)(10,14)(11,12)(16,20)(17,21), ( 1,19)( 2,21)
> ( 3,15)( 4,20)( 5,14)( 6,13)( 7,17)(11,16)(12,18) )
> gap> hp;
> Subgroup( Group( ( 1, 7,12,16,19,21, 6)( 2, 8,13,17,20, 5,11)( 3, 9,14,18, 4,
>  10,15), ( 1, 2)( 3, 6)( 8,15)( 9,13)(10,14)(11,12)(16,20)(17,21), ( 1,19)
> ( 2,21)( 3,15)( 4,20)( 5,14)( 6,13)( 7,17)(11,16)(12,18) ), 
> [ ( 1,15,16, 5,11,13,17)( 2,18, 4,21, 9,14, 8)( 3, 6,20,19,10, 7,12), (), 
>   ( 2, 9)( 3,20)( 5,11)( 7,10)( 8,14)(12,19)(13,16)(15,17)(18,21), 
>   ( 1, 3,21,11,12,14)( 2, 5, 6,18,15, 7)( 4,17,20, 8,13,10)( 9,16,19) ] )
> gap> quit;
> 
> 
> Peter
 
This bug does  not occur any more in the present GAP 3.1 and will also
not  occur in GAP  3.2  (to  be  released  *soon*!!!).  A  bug in  the
normalizer  program was  fixed  in  patchlevel  2  for  3.1,  which is
available since June  2, 1992. So the reported  bug refers most likely
to a copy of GAP that has not been corrected with the  above-mentioned
patch.

Joachim Neubueser



From dana@bimacs.cs.biu.ac.il Wed Feb 10 15:09:43 1993
From:       dana@bimacs.cs.biu.ac.il "Dana-Picard Noah"
Date:       Wed, 10 Feb 93 15:09:43 +0100
Subject:    products

I'm a very new user of GAP and try to run it on 486.
1) If H and K are two subgroups of G, what is the easiest way of
computing their product HK as a subgroup of G?
2) Does it exist some kind of tutorial for the beginner?
Thanks a lot,
Thierry Dana-Picard.



From neubuese@samson.math.rwth-aachen.de Wed Feb 10 17:00:50 1993
From:       neubuese@samson.math.rwth-aachen.de "Joachim Neubueser"
Date:       Wed, 10 Feb 93 17:00:50 +0100
Subject:    Re: products

Let me answer your questions:
> Subject: products
> 
> I'm a very new user of GAP and try to run it on 486.
> 1) If H and K are two subgroups of G, what is the easiest way of
> computing their product HK as a subgroup of G?
> 2) Does it exist some kind of tutorial for the beginner?
> Thanks a lot,
> Thierry Dana-Picard.

ad 1):  Usually  the notation HK for two  subgroups H and K is used to
mean the *set* of all  elements hk with h in H and  k in K.  This is a
subgroup only  if  HK  = KH.  If  this is the  case,  then the command
Closure(H,K) will compute this *subgroup* HK,  as described in section
7.17 (pages 227/8) of the manual. If HK is *not* a subgroup ( as would
alredy happen  if you  take  two  cyclic subgroups  of order 2  in the
symmetric group of degree 3) then there is *no* GAP command to compute
this set, you would have to create this *set* by a little self-written
function.

ad 2): There is  no special tutorial for GAP, however chapter 1 of the
manual,  called  "About   GAP"  (pages   37  -150)  provides  an  easy
introduction  to GAP.  Further  you may be  interested  that  a Summer
School on Computational Group  Theory using GAP as the main vehicle is
planned as part of the  'Groups 1993, Galway/St.Andrews' meeting to be
held in Galway, Ireland, August 1 to 14.

Joachim Neubueser



From caranti@volterra.cineca.it Thu Feb 11 16:00:58 1993
From:       caranti@volterra.cineca.it "Andrea Caranti"
Date:       Thu, 11 Feb 93 16:00:58 +0100
Subject:    combinat.tst

Dear gap-forum,
I just got a new Sun SparcStation 10/41, with 64MB of RAM, running under
Solaris 1.1 at 40MHz. I ran combinat.tst with the standard 4MB and got
exactly 50000 GAPstones for the first run, and 50787 for the second
one. I was using the pre-compiled GAP kernel from Aachen. I will try
to recompile GAP on the machine, and see what I get.

Yours,

A Caranti



From L.H.Soicher@qmw.ac.uk Thu Feb 11 18:31:49 1993
From:       L.H.Soicher@qmw.ac.uk "Leonard Soicher"
Date:       Thu, 11 Feb 93 18:31:49 +0100
Subject:    Blist functions

The functions  UnionBlist,  IntersectionBlist,  and  DifferenceBlist
dont't appear to be defined!

L.H.Soicher@qmw.ac.uk



From martin@bert.math.rwth-aachen.de Thu Feb 11 18:39:54 1993
From:       martin@bert.math.rwth-aachen.de "Martin Schoenert"
Date:       Thu, 11 Feb 93 18:39:54 +0100
Subject:    Re: Blist functions

'UnionBlist' etc. are defined in GAP 3.2.  But 'UniteBlist' etc. are
usually better (because they don't allocate new objects).

Thanks, Martin.

-- .- .-. - .. -.  .-.. --- ...- . ...  .- -. -. .. -.- .-
Martin Sch"onert,   Martin.Schoenert@Math.RWTH-Aachen.DE,  +49 241 804551
Lehrstuhl D f"ur Mathematik, Templergraben 64, RWTH, D 51 Aachen, Germany



From michel@dmi.ens.fr Fri Feb 12 13:13:17 1993
From:       michel@dmi.ens.fr "Jean Michel"
Date:       Fri, 12 Feb 93 13:13:17 +0100
Subject:    problem with empties in GAP

I hope it is not too late to ask for some small fixes for what I consider to be
a bug in the design of some gap functions. To show why I consider this feature to be a bug,
I will show some examples of programs where I had to add ugly code to get around
it -- then you can argue against me constructively by showing me how I should have
written my code.

Example 1:
  Given a list l1,..lr of integers (with no holes -- recognized as a vector by Gap)
I want to construct the list (1,..,d,l1+d,...,lr+d).
So I wrote:

  gap>shift:=function(l,d)return Concatenation([1..d],l+d);end;

  What is the problem? This does not work when l is empty: I get 

  "Error, Vectors: '+' incompatible types"

  Indeed, when asked:

  gap>IsVector([]);
  false

  Is not this wrong? The vector space of dimension 0 is a perfectly valid
  mathematical object -- How to represent its elements?

  I had to write:

  plus:=function(a,b)
   if Length(a)=0 then return a;
   elif Length(b)=0 then return b;
   else return a+b;
   fi;
   end;

  and use it in many places instead of '+' (and similarly for '-').
  Is not this ugly (and inefficient)?

Example 2:
  I need to compute an echelonized basis of the space generated by a set of vectors.
  If s is this set (represented as a list of vectors), TriangulizeMat(s) works well
  excepted when the set is empty:

  gap>TriangulizeMat([]);
  Error, ... because in the code of the function Length(mat[1]) is taken.

  Would not it be easy to fix it so that TriangulizeMat([])=[] ?

Discussion:
  these 2 examples seem to me a symptom of a wrong treatment of empties in GAP, which
  forces to add ugly and unnecessary code to handle special cases. GAP has only one
  kind of empty, the list [] (in contrast to languages like APL where an empty matrix still
  may have a shape like [0,5]). This is fine with me, but then all operations which logically
  should accept empty vectors(or matrices, or whatever...) should also accept [].
  I cannot see any big problem with IsVector([]) being true, but it being false certainly gives me
  problems.

       Jean MICHEL, D.M.I., E.N.S - Paris



From martin@bert.math.rwth-aachen.de Fri Feb 12 17:48:24 1993
From:       martin@bert.math.rwth-aachen.de "Martin Schoenert"
Date:       Fri, 12 Feb 93 17:48:24 +0100
Subject:    Re: problem with empties in GAP

In his e-mail of 1993/02/12 (actually it arrived here a little bit
earlier, but not from his subscription address, so I had to stuff it into
'listserv' by hand) Jean Michel asks why GAP doesn't allow vectors of
length zero.

He writes:

    Given a list l1,..lr of integers (with no holes -- recognized as a
    vector by Gap).  I want to construct the list (1,..,d,l1+d,...,lr+d).
    So I wrote:

        gap>shift:=function(l,d)return Concatenation([1..d],l+d);end;

    What is the problem? This does not work when l is empty: I get 

        "Error, Vectors: '+' incompatible types"

Actually this  will work in GAP 3.2, but it probably should not be relied
upon to work in later releases.

He continues

    Indeed, when asked:

        gap>IsVector([]);
        false

    Is not this wrong? The vector space of dimension 0 is a perfectly valid
    mathematical object -- How to represent its elements?

I beg to differ.  The vector space of dimension 0 *over a given field* is
a perfectly valid object.  But if we represent vectors in such a vector
space by lists of length zero we have way to find the field this vector
lies in.

Why is this a problem?  Well, what is the scalar product of two empty
vectors?  Zero, of course.  But which zero?  The integer zero, or the
zero from a finite field (but of which characteristic), or the zero
polynomial over some ring?  This is basically the problem.  We have no
way to find a field for such an empty vector (and thus the zero).

He contiunes:

    Example 2:

    I need to compute an echelonized basis of the space generated by a set of
    vectors.  If s is this set (represented as a list of vectors),
    TriangulizeMat(s) works well  excepted when the set is empty:

        gap>TriangulizeMat([]);
        Error, ... because in the code of the function Length(mat[1]) is taken.

    Would not it be easy to fix it so that TriangulizeMat([])=[] ?

Yes, it would be easy to fix 'TriangulizeMat', and I see  no problem with
this.  Expect this to happen in the near future.

He continues:

    these 2 examples seem to me a symptom of a wrong treatment of empties in
    GAP, which forces to add ugly and unnecessary code to handle special
    cases. GAP has only one kind of empty, the list [] (in contrast to
    languages like APL where an empty matrix still may have a shape like
    [0,5]). This is fine with me, but then all operations which logically
    should accept empty vectors(or matrices, or whatever...) should also
    accept [].

I would formulate this slightly different (though I agree that there is
a problem).  In GAP certain information about a vector or a matrix
is simply implicit.  For example with vectors the field over which this
vector lies is derived from the entries of this vector.  This is why a
vector must have at least one entry.  Another example are matrices.
The shape of a matrix is derived from its length and the length of its
rows.  Lets for the moment assume that vectors of length zero were legal.
Then we could create a matrix of shape [5,0], namely [[],[],[],[],[]].
But we could not create a matrix of shape [0,5], because how could GAP
guess that each row has length 5, if the matrix has no rows.

Because GAP tries to read of information (the field and the dimension)
about a vector or a matrix from the entries (the elements or the rows)
such a field or matrix may not have zero length.

In spite of this problem I still think that GAP's approach is a good one.
In GAP 2.4 vectors and matrices were different datatypes, and the
neccessary conversions between lists (of lists) and vectors or matrices
were quite a pain.  In GAP 3, where everything is just a list, you have
all the powerfull list operations and functions available to apply to
vectors, matrices (and sets too).  You can simple extract elements from a
matrix with m[i][j] or change them with an assignment (which you could
not in GAP 2.4).

He continues:

    I cannot see any big problem with IsVector([]) being true,
    but it being false certainly gives me problems.

I hope I could convince that there are problems.

Martin.

-- .- .-. - .. -.  .-.. --- ...- . ...  .- -. -. .. -.- .-
Martin Sch"onert,   Martin.Schoenert@Math.RWTH-Aachen.DE,  +49 241 804551
Lehrstuhl D f"ur Mathematik, Templergraben 64, RWTH, D 51 Aachen, Germany



From borbor@divsun.unige.ch Sat Feb 13 07:48:28 1993
From:       borbor@divsun.unige.ch "Borcic Boris"
Date:       Sat, 13 Feb 93 07:48:28 +0100
Subject:    GAP for MAC, new version ?

I think I recall a new version of GAP for the Macintosh
was promised maybe mid-december for maybe mid-january
by the authors of the original port.

Is it still in the works ?

Regards,

Boris Borcic
---
(12^2-55)^2-12^2*55-1=0



From smith@pell.anu.edu.au Mon Feb 15 05:00:32 1993
From:       smith@pell.anu.edu.au "Michael Smith"
Date:       Mon, 15 Feb 93 05:00:32 +0100
Subject:    Running GAP in Emacs buffer.

GAP and Emacs users:
I have just finished writing a GAP process mode for running an interactive
GAP session in an Emacs buffer. This code is based on the GAP mode of Goetz
Pfeiffer, which I modified to use comint-mode instead of shell-mode.

GAP's command completion and help are available.  Both the help and lists
of completions are shown in separate buffers from the GAP session, and help
can be evoked at any time (on any topic, defaulting to current command) by
pressing "?".

The file gap-process.el is available by anonymous ftp from pell.anu.edu.au
in directory pub/gnu/elisp.  Note that it requires the comint-mode package
by Olin Shivers to work.

Follow the instructions in the comments at the start of the file for
installation instructions.

Cheers,
Michael.



From michel@dmi.ens.fr Mon Feb 15 10:37:30 1993
From:       michel@dmi.ens.fr "Jean Michel"
Date:       Mon, 15 Feb 93 10:37:30 +0100
Subject:    Re: problem with empties in GAP

Excuse me for this long message -- I reply to the reply to my message,
and to make things celar I quote extensively both.

I said:

>>> I hope it is not too late to ask for some small fixes for what I consider
>>> to be a bug in the design of some gap functions. To show why I consider this
>>> feature to be a bug, I will show some examples of programs where I had to
>>> add ugly code to get around it -- then you can argue against me
>>> constructively by showing me how I should have written my code.

>>> Example 1:
>>>  Given a list l1,..lr of integers (with no holes -- recognized as a vector by
>>>  Gap) I want to construct the list (1,..,d,l1+d,...,lr+d).
>>>  So I wrote:
>>>
>>>  gap>shift:=function(l,d)return Concatenation([1..d],l+d);end;
>>>
>>>  What is the problem? This does not work when l is empty: I get 
>>>
>>>  "Error, Vectors: '+' incompatible types"
>>>

Martin Schoenert says:
>> Actually this  will work in GAP 3.2, but it probably should not be relied
>> upon to work in later releases.
>> 
What does this mean?

>>>  Indeed, when asked:
>>>
>>>  gap>IsVector([]);
>>>  false
>>>
>>>  Is not this wrong? The vector space of dimension 0 is a perfectly valid
>>>  mathematical object -- How to represent its elements?
>>>

Martin says
>> I beg to differ.  The vector space of dimension 0 *over a given field* is
>> a perfectly valid object.  But if we represent vectors in such a vector
>> space by lists of length zero we have way to find the field this vector
>> lies in.
>> 
>> Why is this a problem?  Well, what is the scalar product of two empty
>> vectors?  Zero, of course.  But which zero?  The integer zero, or the
>> zero from a finite field (but of which characteristic), or the zero
>> polynomial over some ring?  This is basically the problem.  We have no
>> way to find a field for such an empty vector (and thus the zero).
>> 

This answer seems to me unnecessarily confused. It seems to reinforce my
point that the design of GAP in this area has not been well thought
out. It is possible to adopt different viewpoints of the situation:

 -pure object-oriented: everything has a type, then a vector is
 necessarily a vector of something, and Martin's criticism is fully
 valid; also the design of GAP is then terribly flawed since this type
 is not explicitely present in the object vector, which is thus mostly
 unusable.

 -functional: a unique data model (list) serves a variety of purposes.
 Type information is given by context. This is the viewpoint I have
 adopted to try to make sense of the design of GAP. Then the problem
 raised by Martin does not exist: we don't have to find the type of
 an empty object, but the type the context requires, if any:
 e.g.:
 - add a (possibly empty) vector to a number:
    if the vector is nonempty, use the type of the vector's elements
    otherwise that of the number.
 - scalar product of two empty vectors: no type information given by
   context, it is reasonable to give an error. It would be very useful,
   though, to have as answer an object which could serve as zero in a
   variety of fields and rings.
 - TriangulizeMat([]):
    no type information given by context, but none required either,
    since result is empty.
 - And now the big one:
    IsVector([]):
    The context clearly requires 'true' !



From neubuese@samson.math.rwth-aachen.de Mon Feb 15 17:31:02 1993
From:       neubuese@samson.math.rwth-aachen.de "Joachim Neubueser"
Date:       Mon, 15 Feb 93 17:31:02 +0100
Subject:    Re: problem with empties in GAP

Dear Professor Michel,
I  am  answering your letter of  February the 15.th to  the GAP  forum
since Martin Schoenert is absent from  Aachen for the next two  weeks.
Let me first summarize the point of the discussion as I see it, namely
the question in how far GAP supports the interpretation of empty lists
as vectors and allows typical vector  operations with them. At the end
of today's letter you offer two different viewpoints of the situation:

First the  "pure object-oriented: everything has a type ...".  This is
clearly not the viewpoint adopted in GAP 3.1 for the interpretation of
lists as vectors, as Martin  Schoenert explained to you in his letter.
Such a  puristic  viewpoint had been tried  in  GAP  2.4  and has been
discarded  and  removed  in the transition  from  2.4 to  3.0 since we
learned by experience that it created a lot of practical problems.  In
so far it is unnecessery to state that if this was the viewpoint taken
then the design of GAP would be terrible flawed. This viewpoint is not
taken, you know it and Martin said it.

Second the "functional: a unique  data model (list) serves a varity of
purposes...". This is indeed the viewpoint presently taken in GAP, and
again Martin has explained this. Also indeed in a number of cases, GAP
does try to use  the context  in order to interpret what  an operation
should mean in situations where, from a strict  mathematical point  of
view, the operation  is not  defined.  For instance,  if  you  want to
multiply an  integer by a  finite  field element, GAP  does  return  a
finite field element although, strictly speaking, the  two factors are
from  different  domains.  All you seem to  ask for is an extension of
the  ability of GAP  to "guess" from the context what is  meant.  That
this is  indeed  all you ask for is demonstrated by the fact that  you
are willing  to allow an  error message in case of a scalar product of
two "empty vectors".  GAP did return an error message in a place where
it  disturbed you,  but in  principle the  two  places aren't all that
different.

Such request as  shown by the problems that you mention in  your first
letter is certainly reasonable  and ought to be discussed  if problems
of the commodity of using GAP have arisen as it has been the case with
you.  I do not think, however, that such  a request is a reason to use
in a  public worldwide discussion terms like you have  chosen in  your
letters,  like  talking  of "a  wrong  treatment  of  empties" or  "an
unnecessary confused answer" and the like.  While we are grateful  for
each notification  of  a  real bug,  and  while  we  appreciate  every
suggestion for the improvement of GAP, I am not willing to accept that
users of GAP use a tone  in the discussion  with us  like an impatient
teacher may (but should not) use with a dumb student.

In  detail: Martin's  answer  that  adding an integer to an empty list
will work  in GAP  3.2 but one  should not rely upon this  to work  in
later releases  means: This possibility --which will not be  mentioned
in the manual of GAP 3.2-- exists in GAP 3.2 but  the rather difficult
question how to treat such border cases as addressed by you under  the
present viewpoint  of GAP  in more generality may  find another answer
later, forced upon by further experience.

As to the possibility  of doing the  kind of operations you would like
to have with empty vectors: GAP offers you to define a new  data  type
"vector over a  fixed field and of fixed dimension" using records in a
similar way to  the  one explained in the section "About Defining  new
Group  Elements".  Of course some  functions  have  to be written  for
that, and also it has to be expected that operations for this new type
will be less efficient.  A student of Professor Schoenwaelder here has
indeed  experimented with  this in a similar  case  in order to  avoid
difficulties with the start of a recursion.

Sincerely yours
Joachim Neubueser.



From michel@dmi.ens.fr Tue Feb 16 10:33:35 1993
From:       michel@dmi.ens.fr "Jean Michel"
Date:       Tue, 16 Feb 93 10:33:35 +0100
Subject:    Re: apologies

>> Such request as  shown by the problems that you mention in  your first
>> letter is certainly reasonable  and ought to be discussed  if problems
>> of the commodity of using GAP have arisen as it has been the case with
>> you.  I do not think, however, that such  a request is a reason to use
>> in a  public worldwide discussion terms like you have  chosen in  your
>> letters,  like  talking  of "a  wrong  treatment  of  empties" or  "an
>> unnecessary confused answer" and the like.  While we are grateful  for
>> each notification  of  a  real bug,  and  while  we  appreciate  every
>> suggestion for the improvement of GAP, I am not willing to accept that
>> users of GAP use a tone  in the discussion  with us  like an impatient
>> teacher may (but should not) use with a dumb student.
>> 
I wish to apologize for the tone of my replies. I tend to write as I
speak in the heat of the discussion, and always forget that the written
word does not carry any of the verbal and gestual clues which put things
in the proper context. But, please, do not think this reflect an
inferior opinion of your ideas, on the contrary, I only become excited
when discussing exciting topics (and never ever when I speak with
someone dumb).

P.S. I also think now from your reply that my question has been fully
understood, so I wait eagerly for what you will do.

Now another small tought about GAP to make this letter a better fit for
the forum:

I have had often to write code, like: 'find all rows of a matrix where an
entry is the single non-zero in its column' where it would be very
useful to interpret boolean true as 1 and boolean false as 0, like in C:
then this particular code could be:

  Filtered(mat,List(Sum(mat,x->x<>0),y->y=1));

Actually implicit conversion is not necessary, it would be just
as good to write

  Filtered(mat,List(Sum(mat,x->Int(x<>0)),y->y=1));

if Int did the conversion job. And further, a change to GAP is
unnecessary, I can always define:

  BooltoInt:=function(b) if b then return 1 else return 0 fi;end;

But, here is my problem: since GAP is yet interpreted, not compiled,
calling a user-defined function like this is very costly. So what to do:

- can this functionality be added to Int or any other function?
- will it be possible to link to GAP functions written in C?
- will there be a compiler some day?


Best regards, Jean MICHEL



From neubuese@samson.math.rwth-aachen.de Tue Feb 16 17:13:35 1993
From:       neubuese@samson.math.rwth-aachen.de "Joachim Neubueser"
Date:       Tue, 16 Feb 93 17:13:35 +0100
Subject:    Re: your questions

Dear Professor Michel,
thank  you for  your  letter and for understanding my general point on
the discussion.

Let me first explain the state of affairs with the new version of GAP:
I have of course said for a long while that it's release is very close
but this  is now really the  case,  in  fact Thomas Breuer  and  Frank
Celler are just fixing some last known minor  bugs and inconsistencies
that were detected in running through all the examples in  the manual.
And we hope indeed that within this week GAP 3.2 will now be available
through ftp.   This means in particular  that no  major changes can be
made any more for 3.2.

Let me then come back  to the  four detailed questions at  the  end of
your letter of yesterday.

1) adding (or in fact multiplying) the elements of  a list by a number
works  also for  the empty list and returns  the empty list.  As said,
however,  this will not be part of the description in  the  manual for
the  reasons  that I explained in my last letter,  namely,  that  this
whole area  of  interpreting empty  lists  may  at  a  later stage get
revisited.

2) Thomas Breuer has  changed  'TriangulizeMat' and 'BaseMat' to  work
also for empty lists.

3) However, 'IsVector([])'  will still return 'false' for the  reasons
explained in Martin's letter.   The  problem of  changing  this  is  a
nontrivial one and just  changing this  but  then not allowing to form
the  scalar product of two empty lists  which  are recognized as legal
vectors  might confuse users  even more.  We certainly will keep  this
whole area in mind but I cannot promise that we will come up soon with
a neat solution.

Then to the questions that you asked in today's letter:

1) As you say yourself you can  make a  function 'IntBool' that allows
you to filter out all rows  of a matrix  where  an  entry  is a single
nonzero in its column. But indeed, as you say, this will not be highly
efficient. It will probably be much more efficient to write a function
that will  run  through the matrix finding  the columns with  only one
nonzero  entry.   The  reason is  that these one-line statements using
'Filtered'  and 'List' are  elegant but tend to use  too many function
calls.  As you said, function calls to GAP  functions are costly,  but
even if there was an internal function 'IntBool' doing the boolean  to
integer conversion,  the  overhead  created  by 'Filtered', 'List' and
'Sum' would probably outweight this advantage.

2)  At  present  we  certainly  are not  installing  such  a  function
'IntBool' to the kernel and  generally  speaking we hesitate to extend
the kernel unless a proven reasonable gain in efficiency forces  us to
do that.

3) The question of providing means  of linking functions written in  C
to GAP as well as  the question  of a compiler for GAP  have certainly
been discussed here and are  on the list of "dreams" for  GAP. In fact
with  respect  to  a compiler even some  first experiments  have  been
started by a student.   The linking of C functions has  been discussed
also  with  groups  outside Aachen and  if  there  are  very  definite
proposals  that people  might have  we  certainly  like to know  them.
However, both areas are nontrivial tasks and I wouldn't want to commit
myself or anybody else to  promise anything  along these  lines in the
nearer future.

With kind regards
  Joachim Neubueser



From werner@pell.anu.edu.au Tue Feb 16 22:53:33 1993
From:       werner@pell.anu.edu.au "Werner Nickel"
Date:       Tue, 16 Feb 93 22:53:33 +0100
Subject:    integer vector mod integer

Dear GAP forum,

in writing some routines to deal with integer vectors modulo a positive
integer I discovered that the operation   vec mod scalar   in GAP 3.1 is
not possible:

gap> [1,2,3,4,5,6] mod 2;
Error, operations: remainder of list and integer is not defined

It is possible to replace the statement above by the following

gap> List( [1,2,3,4,5,6], x -> x mod 2 );

but at a cost. The following two statements were executed on a Sparc
station SLC.

gap> Runtime();; List( [1..60000], x -> x*2 );; Runtime()-last2;;
10167
gap> Runtime();; [1..60000]*2;; Runtime()-last2;;
1483

It would be useful to have the operation   vec mod scaler   available
for those cases where taking the remainder makes sense. For a similar
reason the operation   vec mod vec   for integer vectors is useful.
The operation would be defined componentwise. In this way one could
easily do computations in finite abelian groups. For example, adding
two vectors in the group C_2 x C_4 x C_12 could be done as follows:

gap> ([1,2,3] + [4,5,6]) mod [2,4,12];
[1,3,9]

With kind regards, Werner Nickel.

----------------------------/|-|--|-|--|------Werner-Nickel-------------------
werner@pell.anu.edu.au     /_| |\ | |  |      Mathematics Research Section
--------------------------/--|-|-\|-|_/|------Australian-National-University--



From neubuese@samson.math.rwth-aachen.de Wed Feb 17 10:06:47 1993
From:       neubuese@samson.math.rwth-aachen.de "Joachim Neubueser"
Date:       Wed, 17 Feb 93 10:06:47 +0100
Subject:    Re: integer vector mod integer

Lieber Herr Nickel,
Thanks for the proposal. It looks nice and worthwhile  and will be put
on the list of quite a few that we have for future releases, but since
realizing it would mean adding to  the  kernel of GAP it is definitely
too late for 3.2.

Joachim Neubueser



From ffor@gauss.math.rochester.edu Wed Feb 17 19:10:34 1993
From:       ffor@gauss.math.rochester.edu "Frederick Ford"
Date:       Wed, 17 Feb 93 19:10:34 +0100
Subject:    GAP on super/parallel computers

I have an opportunity to get some "free" time on either a super  
computer or a parallel computer. Basically, the local administrator  
is getting complaints from users about the lack of speed recently and  
I am currently the most active user on the system. He's hoping that  
getting me off the system will give him some breathing room. I've  
been running very cpu intensive GAP computations.

The super computer options are:

                  OS              C compiler
  IBM ES-9000     AIX/370         AIX/370 C compiler
  IBM RS-6000     AIX             AIX XL C compiler/6000
  Cray-YMP        UNICOS          Cray standard C compiler


The parallel computer is a Connection Machine (massively parallel I'm  
told). The OS is System V, BSD 4.3 compatible. I don't have any  
details on the C compiler version, but it's whatever Thinking  
Machines is distributing as their "standard" C compiler.

I have three questions.

1) The manual indicates that GAP compiles on the IBM RS-6000. Does  
anyone know about any of the other platforms?

2) How much faster, generally speaking, should I expect GAP to be on  
these platforms?

3) Should I opt for super or parallel computing?


Thanks,
Frederick Ford



From jcbhrb@cerf.net Thu Feb 18 00:42:25 1993
From:       jcbhrb@cerf.net "Jacob Hirbawi"
Date:       Thu, 18 Feb 93 00:42:25 +0100
Subject:    Class Multiplication Constants and Character claculation in GAP

gap-forum@samson.math.rwth-aachen.de
One thing that I wish GAP could do is the direct calculation of character
tables, or better yet a full set of irreducible representation matrices.
I also could not find a routine for calculating class multiplication
constants (or coefficients) from the conjugacy classes of group elements;
this would probably be my starting point in calculating characters; oddly
enough there is a routine that uses the character tables to get the class
multiplication constants but that's going in the opposite direction of what
I have in mind. Anyway here are my questions regarding this issue:

  (1) Will version 3.2 have any of the above? or should I consider
      writing my own routines with my very limitted expertise in
      gap?

  (2) If any other user has looked at this before, please pass along
      any experiences or suggestions.

  (3) As an interim solution, is there a way to identify a user defined
      group with a group which the character tables recognize:

      For example here's a group of order 48 which the character tables
      should know:

      a := AbstractGenerator("a");
      b := AbstractGenerator("b");
      c := AbstractGenerator("c");
      z := AbstractGenerator("z");
      group4a := Group(a,b,c,z);
      group4a.relators := [a^2*z^-1,b^3*z^-1,c^4*z^-1,a*b*c*z^-1,z^2];
      group4p:=OperationCosetsFpGroup(group4a,Subgroup(group4a,[IdWord]));

      How do identify group4p to CharTable. (PS. I happen to know for this
      particular case but in general is this doable?)

  (4) When will the next version be released?
      

Any help is greatly appreciated.

Jacob Hirbawi
JcbHrb@CERF.net



From jcbhrb@cerf.net Thu Feb 18 07:03:22 1993
From:       jcbhrb@cerf.net "Jacob Hirbawi"
Date:       Thu, 18 Feb 93 07:03:22 +0100
Subject:    mistake in the characters of 2.S4 ?

gap-forum@samson.math.rwth-aachen.de
While we're on the subject of group characters; there seems to be
a mistake in the characters of 2.S4 . In case anyone is wondering
this all relates to John McKay's recent post on sci.math regarding
the link between the finite dicyclic groups <2,3,3>=SL(2,3), 
<2,3,4>=2.S4, <2,3,5>=2.A5 and the exceptional Lie algebras E6,E7,E8.

gap> CharTable("2.S4");
  ....
 irreducibles := 
 [ [ 1, 1, 1, 1, 1, 1, 1, 1 ], [ 1, 1, 1, -1, -1, -1, 1, 1 ], 
   [ 2, 2, 2, 0, 0, 0, -1, -1 ], [ 3, 3, -1, 1, -1, -1, 0, 0 ], 
   [ 4, -4, 0, 0, 0, 0, 1, -1 ], 
   [ 2, -2, 0, 0, E(8)+E(8)^3, -E(8)-E(8)^3, -1, 1 ], 
   [ 2, -2, 0, 0, -E(8)-E(8)^3, E(8)+E(8)^3, -1, 1 ], 
   [ 3, 3, -1, -1, 1, 1, 0, 0 ] ], 
  ....

I could not duplicate John's constructions with the above characters
but using the table below everything seems to fit nicely:

 [ [ 1, 1, 1, 1, 1, 1, 1, 1 ], [ 1, 1, 1, -1, -1, -1, 1, 1 ], 
   [ 2, 2, 2, 0, 0, 0, -1, -1 ], [ 3, 3, -1, 1, -1, -1, 0, 0 ], 
   [ 4, -4, 0, 0, 0, 0, 1, -1 ], 
   [ 2, -2, 0, 0, E(8)^3+E(8)^5, -E(8)^3-E(8)^5, -1, 1 ], 
   [ 2, -2, 0, 0, -E(8)^3-E(8)^5, E(8)^3+E(8)^5, -1, 1 ], 
   [ 3, 3, -1, -1, 1, 1, 0, 0 ] ], 

both tables pass the TestCharacterTable tests!

Jacob Hirbawi
JcbHrb@CERF.net



From neubuese@samson.math.rwth-aachen.de Thu Feb 18 10:03:45 1993
From:       neubuese@samson.math.rwth-aachen.de "Joachim Neubueser"
Date:       Thu, 18 Feb 93 10:03:45 +0100
Subject:    Re: GAP on super/parallel computers

Frederick Ford asks about GAP on super/parallel computers. I am  sorry
that  from  Aachen  at present  we  cannot  give  much  advice on  his
questions,  we have  not tried any of the  systems  that are mentioned
here.   I think GAP is used  on an  RS-6000  e.g. at  Essen  , perhaps
Gerhard Schneider  (formerly Essen, now Karlsruhe) can  give advice on
IBMs generally.  I  seem to  remember that the  question  of  GAP on a
supercomputer  has  been  mentioned in  the  connection  of  measuring
GAPstones, but I have not kept  that correspondence.  Perhaps somebody
from the GAP-forum has some experience. Also Martin Schoenert may have
some  information from private correspondence, but he will return from
holidays at the beginning of March only.

Generally I would not  expect too much gain from  a  parallel  machine
since there is nothing in GAP specially supporting parallel computing,
but this is pure (maybe poor) guessing. I you dare to try, we would be
interested to hear of the outcome.

Joachim Neubueser



From sam@ernie.math.rwth-aachen.de Thu Feb 18 13:41:57 1993
From:       sam@ernie.math.rwth-aachen.de "Thomas Breuer"
Date:       Thu, 18 Feb 93 13:41:57 +0100
Subject:    

Dear Mrs. and Mr. Forum,
in his message of 18 Feb 93 Jacob Hirbawi asks some questions
concerning character tables.

> One thing that I wish GAP could do is the direct calculation of character
> tables, or better yet a full set of irreducible representation matrices.

In GAP-3.1 both is possible for finite polycyclic groups $G$ with the
property that there is an abelian normal subgroup $N$ of $G$ such that
the factor group $G/N$ is supersolvable.  The GAP functions in question
are 'MatRepresentationsPGroup' and 'CharTablePGroup'.
The algorithm used is described in

U. Baum. Existenz und effiziente Konstruktion schneller
   Fouriertransformationen "uberaufl"osbarer Gruppen.
   Dissertation, Rheinische Friedrich Wilhelm Universit"at Bonn, 1991.

(An English version of this was also published in 1992, but at the moment
I don't find where.)

Jacob Hirbawi continues:

> I also could not find a routine for calculating class multiplication
> constants (or coefficients) from the conjugacy classes of group elements;
> this would probably be my starting point in calculating characters;

In GAP-3.2 there will be an implementation of the so-called Dixon-Schneider
method for the computation of character tables of finite groups.
This algorithm in fact does compute class multiplication constants,
and from that the irreducible characters.  Alexander Hulpke did this
implementation, and he could compute the tables of some large maximal
subgroups of sporadic simple groups using this program, e.g.,
the 8th maximal subgroup of the Conway group Co1, with structure
$2^{2+12}:(A_8 x S_3)$.

To compute the character table of a given group <G> in GAP-3.2 using the
Dixon-Schneider method, one will just call 'CharTable( <G> )'.  Also it
will be possible to combine this method with character theoretical tools
that were available already in GAP-3.1.
The algorithm is described in

J.D. Dixon.  High speed computations of group characters.
   Num. Math. 10, 446-450

and

G.J.A. Schneider. Dixon's Character Table Algorithm Revisited.
   J. Symbolic Computation (1990) 9, 601-606.

As for computing the representations of arbitrary groups (i.e., groups
that are not of the type required for 'CharTablePGroup'), we have no
algorithm up to now.

Jacob Hirbawi continues:

> oddly enough there is a routine that uses the character tables to get the
> class multiplication constants but that's going in the opposite direction
> of what I have in mind.

Often it is useful to compute class multiplication constants from character
tables, e.g., if the group is too large for computations.  First, the
question whether there is a class C in a group G such that G = CC can be
answered by computing class multiplication constants; for the 26
sporadic simple groups the answer is "yes", this was checked from the
character tables in

J. Neub"user, H. Pahlings, E.Cleuvers.  Each sporadic finasig G has a class
  C such that CC = G.  Abstracts AMS, 6 (34), 1984.

Second, it is often possible to prove that a group is a Galois group over
the Rationals using a theorem of Belyi, Fried, Matzat, and Thompson.
One part is the inspection of class multiplication coefficient, the other
part requires inspection of maximal subgroups of the given group.
Such calculations can be found in

H. Pahlings, Some Sporadic Groups as Galois Groups.
  Rend. Sem. Mat. Univ. Padova, Vol. 79 (1988)

and

H. Pahlings, Some Sporadic Groups as Galois Groups II.
  Rend. Sem. Mat. Univ. Padova, Vol. 82 (1989).


Jacob Hirbawi continues:

>  Anyway here are my questions regarding this issue:
> 
>   (1) Will version 3.2 have any of the above? or should I consider
>       writing my own routines with my very limitted expertise in
>       gap?
> 
>   (2) If any other user has looked at this before, please pass along
>       any experiences or suggestions.
> 
>   (3) As an interim solution, is there a way to identify a user defined
>       group with a group which the character tables recognize:
> 
>       For example here's a group of order 48 which the character tables
>       should know:
> 
>       a := AbstractGenerator("a");
>       b := AbstractGenerator("b");
>       c := AbstractGenerator("c");
>       z := AbstractGenerator("z");
>       group4a := Group(a,b,c,z);
>       group4a.relators := [a^2*z^-1,b^3*z^-1,c^4*z^-1,a*b*c*z^-1,z^2];
>       group4p:=OperationCosetsFpGroup(group4a,Subgroup(group4a,[IdWord]));
> 
>       How do identify group4p to CharTable. (PS. I happen to know for this
>       particular case but in general is this doable?)
> 
>   (4) When will the next version be released?

As already said, the answer to the first part of (1) is "yes".      

Question (3) addresses a more general problem:  In GAP-3.1 there was no
very close connection between character tables and groups, dealing with
character tables was mainly talking about groups, not working with groups.
But of course one wants to deal with both the group and its table in many
situations.  When the table was computed from the group in GAP, there is
no problem, since the ordering of conjugacy classes in table and group are
the same.  But if one has a group, and wants to identify its classes
with the columns of a given library table (e.g., one contained in the
ATLAS of finite groups), this is not supported by GAP.  For some series
of groups there are generic character tables in GAP, and the parameters
allow to identify classes and characters.  In the future these parameters
will be added to the library tables belonging to such a series.

The answer to (4) is "tomorrow".


The second message of Jacob Hirbawi tells about a problem with library
tables.  He writes

> While we're on the subject of group characters; there seems to be
> a mistake in the characters of 2.S4 . In case anyone is wondering
> this all relates to John McKay's recent post on sci.math regarding
> the link between the finite dicyclic groups <2,3,3>=SL(2,3), 
> <2,3,4>=2.S4, <2,3,5>=2.A5 and the exceptional Lie algebras E6,E7,E8.
> 
> gap> CharTable("2.S4");
>   ....
>  irreducibles := 
>  [ [ 1, 1, 1, 1, 1, 1, 1, 1 ], [ 1, 1, 1, -1, -1, -1, 1, 1 ], 
>    [ 2, 2, 2, 0, 0, 0, -1, -1 ], [ 3, 3, -1, 1, -1, -1, 0, 0 ], 
>    [ 4, -4, 0, 0, 0, 0, 1, -1 ], 
>    [ 2, -2, 0, 0, E(8)+E(8)^3, -E(8)-E(8)^3, -1, 1 ], 
>    [ 2, -2, 0, 0, -E(8)-E(8)^3, E(8)+E(8)^3, -1, 1 ], 
>    [ 3, 3, -1, -1, 1, 1, 0, 0 ] ], 
>   ....
> 
> I could not duplicate John's constructions with the above characters
> but using the table below everything seems to fit nicely:
> 
>  [ [ 1, 1, 1, 1, 1, 1, 1, 1 ], [ 1, 1, 1, -1, -1, -1, 1, 1 ], 
>    [ 2, 2, 2, 0, 0, 0, -1, -1 ], [ 3, 3, -1, 1, -1, -1, 0, 0 ], 
>    [ 4, -4, 0, 0, 0, 0, 1, -1 ], 
>    [ 2, -2, 0, 0, E(8)^3+E(8)^5, -E(8)^3-E(8)^5, -1, 1 ], 
>    [ 2, -2, 0, 0, -E(8)^3-E(8)^5, E(8)^3+E(8)^5, -1, 1 ], 
>    [ 3, 3, -1, -1, 1, 1, 0, 0 ] ], 
> 
> both tables pass the TestCharacterTable tests!

Both lists of irreducible characters belong to character tables of groups
of structure 2.S4, so orthogonality relations are satisfied.  The table with
name "2.S4" in the table library is one of them, namely the table of the
involution centralizer in the Mathieu group M11.  The other table is that
of an isoclinic group, which can be got in GAP using the following
commmands.

      gap> t:= CharTable( "2.S4" );;
      gap> DisplayCharTable( t );
      2.S4
      
         2  4  4  3  2   3   3  1  1
         3  1  1  .  .   .   .  1  1
      
           1a 2a 4a 2b  8a  8b 3a 6a
        2P 1a 1a 2a 1a  4a  4a 3a 3a
        3P 1a 2a 4a 2b  8a  8b 1a 2a
      
      X.1   1  1  1  1   1   1  1  1
      X.2   1  1  1 -1  -1  -1  1  1
      X.3   2  2  2  .   .   . -1 -1
      X.4   3  3 -1  1  -1  -1  .  .
      X.5   4 -4  .  .   .   .  1 -1
      X.6   2 -2  .  .   A  -A -1  1
      X.7   2 -2  .  .  -A   A -1  1
      X.8   3  3 -1 -1   1   1  .  .
      
      A = E(8)+E(8)^3
        = ER(-2) = i2
      gap> iso:= CharTableIsoclinic( t );;
      gap> DisplayCharTable( iso );
      Isoclinic(2.S4)
      
         2  4  4  3  2   3   3  1  1
         3  1  1  .  .   .   .  1  1
      
           1a 2a 4a 4b  8a  8b 3a 6a
        2P 1a 1a 2a 2a  4a  4a 3a 3a
        3P 1a 2a 4a 4b  8b  8a 1a 2a
      
      X.1   1  1  1  1   1   1  1  1
      X.2   1  1  1 -1  -1  -1  1  1
      X.3   2  2  2  .   .   . -1 -1
      X.4   3  3 -1  1  -1  -1  .  .
      X.5   4 -4  .  .   .   .  1 -1
      X.6   2 -2  .  .   A  -A -1  1
      X.7   2 -2  .  .  -A   A -1  1
      X.8   3  3 -1 -1   1   1  .  .
      
      A = -E(8)+E(8)^3
        = -ER(2) = -r2

At the moment, the function 'CharTableIsoclinic' does this job only for
tables with underlying group of structure 2.G.2, as in this example.
Note that not only the characters may be different but also the power
maps.

Clearly it is a problem to access tables via names that do not uniquely
determine the group but this is the most convenient way to deal with the
tables that occur in the ATLAS of finite groups which serves as standard
for the table names.

Best wishes
Thomas Breuer
(sam@ernie.math.rwth-aachen.de)



From neubuese@samson.math.rwth-aachen.de Thu Feb 18 14:42:51 1993
From:       neubuese@samson.math.rwth-aachen.de "Joachim Neubueser"
Date:       Thu, 18 Feb 93 14:42:51 +0100
Subject:    Re: GAP for MAC, new version ?

On Feb. 13, Boris Borcic asked:
> I think I recall a new version of GAP for the Macintosh
> was promised maybe mid-december for maybe mid-january
> by the authors of the original port.
> 
> Is it still in the works ?

I am afraid a lot of promises have been made  about the new version of
GAP  coming  out  soon,  but that they have not been fulfilled is  our
fault,  not  that  of  the  group  of  people  working  with Professor
Mendelsohn in the University of Manitoba who kindly undertook the port
to  MACs.  However  after these days really the  new  version is being
released,  I hope that we will  get again  a  port to  the  MAC family
through the help of the same group of people.

Joachim Neubueser



From sl25@cus.cam.ac.uk Thu Feb 18 17:30:39 1993
From:       sl25@cus.cam.ac.uk "Steve Linton"
Date:       Thu, 18 Feb 93 17:30:39 +0100
Subject:    Re: GAP on super/parallel computers 

Everything I am about to say is guesswork, and I would be delighted
to be proved wrong:

GAP is very unlikely to benefit from a massively parallel system
such as a connection machine without considerable help. In
principle, some (though not very many) of the array operations
involved in large list or permutation calculations could be speeded
up, but they would have to be VERY large before the gain exceeded
the overhead of distributing the data to the various machines. Also
the compiler is unlikely to spot any useful parallelisations in the
GAP kernel. It will be looking for floating point array calculations
and the like.

Super-computers are also usually optimised mainly for floating point
array calculations, but will probably still give you some gain on
large list, permutation and vector operations. They also usually
have basic scalar processors comparable to the fastest workstations,
and lots of real memory, both of which may help you.

Of the options you suggest, I'd recommend the RS/6000, on the
grounds of ease of porting and ease of use. The ES/9000 might be a
few times faster as would the Cray, but you'd really be wasting
their capabilities, and you'd probably have to do some work on the
porting.

I don't know what machine you're working on, but a Sun Sparcstation
1+ scores about 15000 GAPstones (see the archives of this list for
lots more ratings) and an RS6000 (possibly not the model you have in
mind) scores 41000. I would guess (pure stab in the dark) at about
100000 for the ES/9000, and perhaps 200000 on the Cray, but I could
be miles out.

Have you made sure that you can't speed up your programs by
improving algorithms, or even by recoding key steps in C? This is
usually the best choice where it's possible.

	Steve



From dawn@math.wayne.edu Thu Feb 18 22:09:52 1993
From:       dawn@math.wayne.edu "Dawn Endico"
Date:       Thu, 18 Feb 93 22:09:52 +0100
Subject:    Re: GAP on super/parallel computers

>I don't know what machine you're working on, but a Sun Sparcstation
>1+ scores about 15000 GAPstones (see the archives of this list for
>lots more ratings) and an RS6000 (possibly not the model you have in

Archives?!?  That reminds me that I've been meaning to ask whether
anyone has gap-forum archives available on a gohper server.  FTP?
 
--
Dawn Endico                       dawn@math.wayne.edu



From fceller@bert.math.rwth-aachen.de Fri Feb 19 20:11:14 1993
From:       fceller@bert.math.rwth-aachen.de "Frank Celler"
Date:       Fri, 19 Feb 93 20:11:14 +0100
Subject:    GAP 3.2

Dear Forum,
eventually we release GAP  3.2. Please  allow a few days until the FTP
servers outside Aachen mentioned below will provide the new files. The
files  "gapexe.su3",   "gapexe.st"  and  "gapexe.next"   are  not  yet
available but will be added as soon as possible.

Have fun with GAP
  Thomas Breuer, Frank Celler and Alexander Hulpke

-----------------------------------------------------------------------------

Introduction
============

    GAP  is  a  system  for computational  discrete  algebra,  which we  have
    developed  with  particular  emphasis on  computational group theory, but
    which has already proved useful also in  other areas.  The name GAP is an
    acronym for *Groups, Algorithms, and Programming*.  This  (long) document
    announces the availability of GAP version 3 release 2, GAP 3.2 for short.
    It is an *advertisement* for GAP, but not a *commercial*, since  we  give
    GAP away for free.

    This document begins with the section "Announcement", which  contains the
    announcement proper.  The next section "Analyzing Rubik's Cube with  GAP"
    contains  an  extensive example.  This example is  followed by  a general
    discussion  of GAP's  capabilities  in  the section "An Overview of GAP".
    The section  "What's New in 3.2"  tells you about the new features in GAP
    3.2.   The  next  sections  "How  to  get  GAP" and "How  to install GAP"
    describe  how you can get GAP running on your computer.  Then we tell you
    about our plans for the future in the section  "The  Future of GAP".  The
    final section "The GAP Forum" introduces the GAP forum, where  interested
    users can discuss GAP related topics by e-mail messages.


Announcement
============
                                                            Il est trop tard,
                                                                  maintenant,
                                                  il sera toujours trop tard.
                                                                Heureusement!
                                                         (A. Camus, La chute)


                 ########            Lehrstuhl D fuer Mathematik
               ###    ####           RWTH Aachen
              ##         ##
             ##          #             #######            #########
            ##                        #      ##          ## #     ##
            ##           #           #       ##             #      ##
            ####        ##           ##       #             #      ##
             #####     ###           ##      ##             ##    ##
               ######### #            #########             #######
                         #                                  #
                        ##           Version 3              #
                       ###           Release 2              #
                      ## #           12 Feb 93              #
                     ##  #
                    ##   #  Alice Niemeyer, Werner Nickel,  Martin Schoenert
                   ##    #  Johannes Meier, Alex Wegner,    Thomas Bischops
                  ##     #  Frank Celler,   Juergen Mnich,  Udo Polis
                  ###   ##  Thomas Breuer,  Goetz Pfeiffer, Hans U. Besche
                   ######   Volkmar Felsch, Heiko Theissen, Alexander Hulpke
                            Ansgar Kaup,    Akos Seress


    Lehrstuhl D  f"ur Mathematik, RWTH Aachen,  announces the availability of
    GAP version 3  release 2,  or  GAP  3.2  for  short.  This  is the  first
    publicly  available  release  of   GAP  since  version  3.1,  which   was
    distributed since April 1992.


Analyzing Rubik's Cube with GAP
===============================
                                   Ideal Toy Company stated on the package of
                            the original Rubik cube that there were more than
                         three billion possible states the cube could attain.
                            It's analogous to Mac Donald's proudly announcing
                                  that they've sold more than 120 hamburgers.
                                                   (J. A. Paulos, Innumeracy)

    To show you what GAP can do a short example is probably best.  If you are
    not interested in this example skip to the  section "An Overview of GAP".

    For the example we consider the group of transformations of Rubik's magic
    cube.  If we number the faces of this cube as follows

                         +--------------+
                         |  1    2    3 |
                         |  4  top    5 |
                         |  6    7    8 |
          +--------------+--------------+--------------+--------------+
          |  9   10   11 | 17   18   19 | 25   26   27 | 33   34   35 |
          | 12  left  13 | 20 front  21 | 28 right  29 | 36  rear  37 |
          | 14   15   16 | 22   23   24 | 30   31   32 | 38   39   40 |
          +--------------+--------------+--------------+--------------+
                         | 41   42   43 |
                         | 44 bottom 45 |
                         | 46   47   48 |
                         +--------------+

    then the  group is  generated by the following  generators, corresponding
    to the six faces of the cube  (the two  semicolons tell GAP  not to print
    the result, which is identical to the input here).

      gap> cube := Group(
      >   ( 1, 3, 8, 6)( 2, 5, 7, 4)( 9,33,25,17)(10,34,26,18)(11,35,27,19),
      >   ( 9,11,16,14)(10,13,15,12)( 1,17,41,40)( 4,20,44,37)( 6,22,46,35),
      >   (17,19,24,22)(18,21,23,20)( 6,25,43,16)( 7,28,42,13)( 8,30,41,11),
      >   (25,27,32,30)(26,29,31,28)( 3,38,43,19)( 5,36,45,21)( 8,33,48,24),
      >   (33,35,40,38)(34,37,39,36)( 3, 9,46,32)( 2,12,47,29)( 1,14,48,27),
      >   (41,43,48,46)(42,45,47,44)(14,22,30,38)(15,23,31,39)(16,24,32,40)
      > );;

    First we want to know the size of this group.

      gap> Size( cube );
      43252003274489856000

    Since this is a little bit unhandy, let us factorize this number.

      gap> Collected( Factors( last ) );
      [ [ 2, 27 ], [ 3, 14 ], [ 5, 3 ], [ 7, 2 ], [ 11, 1 ] ]

    (The result tells us that the size is 2^27 3^14 5^3 7^2 11.)

    Next let us investigate the operation of the group on the 48 points.

      gap> orbits := Orbits( cube, [1..48] );
      [ [ 1, 3, 17, 14, 8, 38, 9, 41, 19, 48, 22, 6, 30, 33, 43, 11, 46,
            40, 24, 27, 25, 35, 16, 32 ],
        [ 2, 5, 12, 7, 36, 10, 47, 4, 28, 45, 34, 13, 29, 44, 20, 42,
            26, 21, 37, 15, 31, 18, 23, 39 ] ]

    The  first orbit contains the points at the corners, the second  those at
    the edges; clearly the group cannot move a point at a corner onto a point
    at an edge.

    So to  investigate the cube group  we first investigate  the operation on
    the corner points.  Note that  the constructed group that describes  this
    operation  will  operate  on the set   [1..24], not  on the  original set
    [1,3,17,14,8,38,9,41,19,48,22,6,30,33,43,11,46,40,24,27,25,35,16,32].

      gap> cube1 := Operation( cube, orbits[1] );
      Group( ( 1, 2, 5,12)( 3, 7,14,21)( 9,16,22,20),
             ( 1, 3, 8,18)( 4, 7,16,23)(11,17,22,12),
             ( 3, 9,19,11)( 5,13, 8,16)(12,21,15,23),
             ( 2, 6,15, 9)( 5,14,10,19)(13,21,20,24),
             ( 1, 4,10,20)( 2, 7,17,24)( 6,14,22,18),
             ( 4,11,13, 6)( 8,15,10,17)(18,23,19,24) )
      gap> Size( cube1 );
      88179840

    Now this group obviously operates transitively, but let us  test  whether
    it is also primitive.

      gap> corners := Blocks( cube1, [1..24] );
      [ [ 1, 7, 22 ], [ 2, 14, 20 ], [ 3, 12, 16 ], [ 4, 17, 18 ],
        [ 5, 9, 21 ], [ 6, 10, 24 ], [ 8, 11, 23 ], [ 13, 15, 19 ] ]

    Those eight  blocks correspond to  the eight corners of  the cube; on the
    one hand the group permutes those and on the  other hand it  permutes the
    three points at each corner cyclically.

    So the obvious thing to do is to  investigate the operation  of the group
    on the eight corners.

      gap> cube1b := Operation( cube1, corners, OnSets );
      Group( (1,2,5,3), (1,3,7,4), (3,5,8,7),
             (2,6,8,5), (1,4,6,2), (4,7,8,6) )
      gap> Size( cube1b );
      40320

    Now a permutation group of degree 8 that has order 40320 must be the full
    symmetric group S(8) on eight points.

    The next thing then  is to investigate  the  kernel  of this operation on
    blocks, i.e.,  the  subgroup of  'cube1'  of those elements that  fix the
    blocks setwise.

      gap> blockhom1 := OperationHomomorphism( cube1, cube1b );;
      gap> Factors( Size( Kernel( blockhom1 ) ) );
      [ 3, 3, 3, 3, 3, 3, 3 ]
      gap> IsElementaryAbelian( Kernel( blockhom1 ) );
      true

    We can  show that the product of  this elementary  abelian group 3^7 with
    the S(8) is semidirect by finding a complement, i.e., a subgroup that has
    trivial intersection with the kernel  and that generates 'cube1' together
    with the kernel.

      gap> cmpl1 := Stabilizer( cube1, [1,2,3,4,5,6,8,13], OnSets );;
      gap> Size( cmpl1 );
      40320
      gap> Size( Intersection( cmpl1, Kernel( blockhom1 ) ) );
      1
      gap> Closure( cmpl1, Kernel( blockhom1 ) ) = cube1;
      true

    There is even a more elegant way to show that 'cmpl1' is a complement.

      gap> IsIsomorphism( OperationHomomorphism( cmpl1, cube1b ) );
      true

    Of course,  theoretically  it is  clear  that 'cmpl1'   must  indeed be a
    complement.

    In  fact we  know that  'cube1' is a  subgroup of index 3 in  the  wreath
    product of a  cyclic 3 with S(8).  This  missing index 3 tells us that we
    do  not have total freedom in turning the  corners.  The  following tests
    show  that whenever we  turn one  corner clockwise we  must  turn another
    corner counterclockwise.

      gap> (1,7,22) in cube1;
      false
      gap> (1,7,22)(2,20,14) in cube1;
      true

    More or less the same things happen when we consider the operation of the
    cube group on the edges.

      gap> cube2 := Operation( cube, orbits[2] );;
      gap> Size( cube2 );
      980995276800
      gap> edges := Blocks( cube2, [1..24] );
      [ [ 1, 11 ], [ 2, 17 ], [ 3, 19 ], [ 4, 22 ], [ 5, 13 ], [ 6, 8 ],
        [ 7, 24 ], [ 9, 18 ], [ 10, 21 ], [ 12, 15 ], [ 14, 20 ], [ 16, 23 ] ]
      gap> cube2b := Operation( cube2, edges, OnSets );;
      gap> Size( cube2b );
      479001600
      gap> blockhom2 := OperationHomomorphism( cube2, cube2b );;
      gap> Factors( Size( Kernel( blockhom2 ) ) );
      [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ]
      gap> IsElementaryAbelian( Kernel( blockhom2 ) );
      true
      gap> cmpl2 := Stabilizer(cube2,[1,2,3,4,5,6,7,9,10,12,14,16],OnSets);;
      gap> IsIsomorphism( OperationHomomorphism( cmpl2, cube2b ) );
      true

    This time we get a semidirect  product  of a 2^11 with an S(12), namely a
    subgroup of index 2 of the wreath product of a cyclic 2 with S(12).  Here
    the missing index 2 tells us again that we  do not  have total freedom in
    turning the edges.   The following tests show  that whenever  we flip one
    edge we must also flip another edge.

      gap> (1,11) in cube2;
      false
      gap> (1,11)(2,17) in cube2;
      true

    Since 'cube1' and  'cube2' are the groups  describing the actions  on the
    two orbits of 'cube', it is clear  that 'cube'  is a subdirect product of
    those  groups, i.e., a  subgroup of the direct  product.    Comparing the
    sizes  of 'cube1',  'cube2',  and  'cube' we   see that 'cube'  must be a
    subgroup of index 2 in the direct product of those two groups.

      gap> Size( cube );
      43252003274489856000
      gap> Size( cube1 ) * Size( cube2 );
      86504006548979712000

    This final missing index 2 tells us that we cannot operate on corners and
    edges totally independently.  The following  tests show that whenever  we
    exchange a  pair of  corners we must also  exchange a  pair of edges (and
    vice versa).

      gap> (17,19)(11,8)(6,25) in cube;
      false
      gap> (7,28)(18,21) in cube;
      false
      gap> (17,19)(11,8)(6,25)(7,28)(18,21) in cube;
      true

    Finally let us compute the centre of the cube group, i.e.,  the  subgroup
    of those operations that  can be performed  either before  or  after  any
    other operation with the same result.

      gap> Centre( cube );
      Subgroup( cube, [ ( 2,34)( 4,10)( 5,26)( 7,18)(12,37)(13,20)
                        (15,44)(21,28)(23,42)(29,36)(31,45)(39,47) ] )

    We  see that  the centre  contains  one  nontrivial element, namely   the
    operation that flips all 12 edges simultaneously.

    This concludes our  example.  Of course,  GAP  can do much more, and  the
    next section gives an overview   of its capabilities, but   demonstrating
    them all would take too much room.


An Overview of GAP
==================
                                                      Though this be madness,
                                                    yet there is method in't.
                                                     (W. Shakespeare, Hamlet)

    GAP consists of several parts: the kernel, the  library of functions, the
    library of groups and related data, and the documentation.

    The  *kernel* implements an automatic  memory management,  a  PASCAL-like
    programming  language,  also  called  GAP,  with  special  datatypes  for
    computations  in group theory, and an interactive programming environment
    to run programs written in the GAP programming language.

    The automatic  *memory management* allows  programmers  to concentrate on
    implementing the algorithm without  needing  to care about allocation and
    deallocation  of memory.   It   includes   a  garbage   collection   that
    automatically throws away objects that are no longer accessible.

    The GAP programming language supports a number  of datatypes for elements
    of fields.  *Integers* can be  arbitrarily large, and  are implemented in
    such  a  way that operations  with  small   integers are reasonably fast.
    Building on this large-integer  arithmetic GAP supports  *rationals*  and
    elements  from *cyclotomic  fields*.   Also GAP allows  one to  work with
    elements from *finite fields* of size (at present) at most 2^16.

    The  special datatypes of  group elements are *permutations*,  *matrices*
    over the  rationals, cyclotomic  fields, and  finite  fields,  *words  in
    abstract generators*, and *words in solvable groups*.

    GAP also contains  a very flexible  *list* datatype.  A list is  simply a
    collection of objects  that allows you to access  the components using an
    integer position.  Lists grow automatically when you add new elements  to
    them.  Lists are used to represent sets,  vectors, and matrices.  A *set*
    is represented   by a  sorted  list without  duplicates.   A   list whose
    elements all lie in a common field  is a *vector*.  A  list of vectors of
    the same length over a common field is a  *matrix*.  Since sets, vectors,
    and matrices are lists, all list operations and functions are applicable.
    You can, for example, find a certain element in a vector with the general
    function  'Position'.   There    are   also   *ranges*,  i.e., lists   of
    consecutive  integers, and  *boolean lists*,  i.e., lists containing only
    'true'  and 'false'.  Vectors, ranges,   and boolean lists  have  special
    internal representations to ensure efficient operations and memory usage.
    For example, a boolean list requires only one bit per element.

    *Records*   in GAP are   similar  to   lists,  except that accessing  the
    components of a record is done using a name instead of an index.  Records
    are used to collect objects of different types,  while lists usually only
    contain elements of one type.  Records are for example  used to represent
    groups and   other  domains; there  is  *no*  group datatype  in  the GAP
    language .  Because of this all information  that GAP knows about a group
    is also accessible to you by simply investigating the record.

    The control structures of GAP are PASCAL-like.  GAP  has *if* statements,
    *while*,  *repeat*,  and  *for* loops.  The for   loop  is  a little  bit
    uncommon in that it always loops over the elements of a  list.  The usual
    semantics can be obtained by looping over the elements of a range.  Using
    those building   blocks you can  write   *functions*.   Functions  can be
    recursive, and are first class objects in the sense that you  can collect
    functions in  lists, pass them as  arguments to other  functions and also
    return them.

    It  is important to  note  that GAP has dynamic  typing instead of static
    typing.  That means that the datatype is a property of the object, not of
    the variable.  This allows you to  write general  functions.  For example
    the generic  function that computes an orbit  can be  used to compute the
    orbit of  an  integer under a permutation group,  the  orbit of a  vector
    under a matrix group, the conjugacy class of  a group element,  and  many
    more.

    The  kernel also implements an *interactive  environment* that allows you
    to use GAP.  This environment supports debugging;  in case of  an error a
    break loop is entered in which you can investigate the problem, and maybe
    correct it  and continue.   You also have online  access  to the  manual,
    though sections  that  contain larger formulas  do not look nice   on the
    screen.

    The *library of  functions*,  simply called   library  in  the following,
    contains implementations  of various group theoretical algorithms written
    in the GAP language.  Because all the group  theoretical functions are in
    this library it is easy for you to look  at  them  to find  out  how they
    work, and change them if they do almost, but not quite, what you want.

    The whole   library is   centered around the   concept  of   domains  and
    categories.  A *domain* is a structured set, e.g., a group is a domain as
    is the ring of Gaussian integers.  Each  domain in GAP  belongs to one or
    more *categories*, which are simply sets of domains, e.g., the set of all
    groups forms a category.  The categories in which a domain lies determine
    the functions that are applicable to this domain and its elements.

    To each  domain belongs a set of  functions,  in a  so called  operations
    record, that are called by dispatchers  like 'Size'.   For example, for a
    permutation group  <G>, '<G>.operations.Size' is a  function implementing
    the Schreier Sims algorithm.  Thus  if  you have any domain  <D>,  simply
    calling 'Size( <D> )' will return the size of the domain <D>, computed by
    an  appropriate function.   Domains *inherit*  such functions from  their
    category,  unless they redefine  them.   For example,  for  a permutation
    group  <G>,  the derived subgroup will  be  computed by the generic group
    function, which computes the normal closure of  the subgroup generated by
    the commutators of the generators.

    Of course the most important category is the category of *groups*.  There
    are  about  100  functions applicable  to  groups.  These include general
    functions  such  as  'Centralizer'  and  'SylowSubgroup', functions  that
    compute series of subgroups such as 'LowerCentralSeries', a function that
    computes the whole lattice of subgroups, functions  that test  predicates
    such  as 'IsSimple',  functions that  are related  to the  operations  of
    groups  such as 'Stabilizer', and many more.  Most of these functions are
    applicable to all  groups,  e.g., permutation groups,  finite  polycyclic
    groups, factor groups, direct products of  arbitrary groups, and even new
    types of groups that you create by simply specifying how the elements are
    multiplied and inverted (actually  it is not quite so simple, but you can
    do it).

    Where  the general  functions that are applicable to  all  groups are not
    efficient enough,  we  have  tried  to  overlay them  by  more  efficient
    functions for special types of groups.  The prime example is the category
    of   *permutation   groups*,    which   overlays   'Size',    'Elements',
    'Centralizer', 'Normalizer', 'SylowSubgroup', and a few more functions by
    functions  that  employ  stabilizer chains  and  backtracking algorithms.
    Also many  of  the  functions  that  deal with operations  of groups  are
    overlayed for permutation groups for the operation of a permutation group
    on integers or lists of integers.

    Special  functions for *finitely presented  groups* include  functions to
    find the  index of  a  subgroup via a Todd-Coxeter coset  enumeration, to
    compute  the  abelian  invariants of  the  commutator  factor  group,  to
    intersect two subgroups, to find the  normalizer  of a  subgroup, to find
    all  subgroups of  small index, and to compute and simplify presentations
    for  subgroups.   Of  course it is possible  to go to a permutation group
    operating  on  the cosets  of  a subgroup and  then  to  work  with  this
    permutation group.

    For   *finite   polycyclic  groups*   a  special   kind  of  presentation
    corresponding to a  composition  series  is used.   Such  a  presentation
    implies  a  canonical  form  for the  elements  and thus allows efficient
    operations with the elements of such a group.  This  presentation is used
    to make  functions such  as 'Centralizer',  'Normalizer', 'Intersection',
    and  'ConjugacyClasses' very efficient.   GAP's  capabilities for  finite
    polycyclic groups  exceed  those of the computer system SOGOS (which  was
    developed at Lehrstuhl D f"ur Mathematik for the last decade).

    There  is  also  support for  *mappings* and *homomorphisms*.  Since they
    play such a  ubiquitous role in mathematics, it is only natural that they
    should also play an important role  in a system like GAP.   Mappings  and
    homomorphisms  are objects in  their own right in GAP.   You can apply  a
    mapping to an element of its source, multiply mappings (provided that the
    range of the first  is a  subset of  the  source of  the second),  invert
    mappings (even if what you get is  a multi-valued mapping), and perform a
    few more operations.   Important examples are  the  'NaturalHomomorphism'
    onto  a  factor group,  'OperationsHomomorphism'  mapping  a  group  that
    operates  on a set of <n> elements into the symmetric  group on [1..<n>],
    'Embeddings' into  products of groups,  'Projections'  from  products  of
    groups onto the components,  and the  general 'GroupHomomorphismByImages'
    for which you only specify the images of a set of generators.

    The  library contains a  package for handling  character tables of finite
    groups.  This  includes almost  all possibilities of the computer  system
    CAS (which was  developed  at  Lehrstuhl  D  f"ur  Mathematik in the last
    decade), and  many  new functions.  You can  compute character  tables of
    groups, or construct  character  tables  using other  tables, or  do some
    calculations  within  known  character  tables.   You  can, for  example,
    compute a list of candidates for permutation characters.  Of course there
    are  many character tables (at the moment  more than 650 ordinary tables)
    in the data library, including all those in the ATLAS of finite groups.

    For  large integers  we now  also  have a  package for *elementary number
    theory*.  There  are functions in this package to test primality,  factor
    integers of  reasonable  size,  compute the  size phi(<n>)  of the  prime
    residue group modulo an integer <n>, compute roots modulo an integer <n>,
    etc.  Also based on this there  is  a package  to do  calculations in the
    ring of Gaussian integers.

    The library  also includes a package for  *combinatorics*.  This contains
    functions to find all selections of various flavours of the elements of a
    set, e.g., 'Combinations' and 'Tuples', or the number of such selections,
    e.g., 'Binomial'.  Other functions  are related to  partitions of sets or
    integers, e.g., 'PartitionsSet' and 'RestrictedPartitions', or the number
    of   such, e.g., 'NrPartitions'    and  'Bell'.   It  also contains  some
    miscellaneous functions such as 'Fibonacci' and 'Bernoulli'.

    The *data library* at present  contains the primitive  permutation groups
    of degree up to 50  from C.  Sims, the 2-groups of size dividing 256 from
    E. O'Brien  and  M. F. Newman,  the  3-groups of size  dividing  729 from
    E. O'Brien and C. Rhodes,  the solvable  groups  of  size  up to 100 from
    M. Hall, J. K. Senior, R. Laue,  and J. Neub"user, a library of character
    tables including all of the ATLAS, and a library of tables  of  marks for
    various groups.  We plan to extend the data library with more data in the
    future.

    Together with GAP 3.2 we now distribute several *share library packages*.
    Such packages have been contributed  by other authors, but the  copyright
    remains with the author.  Currently there are three packages in the share
    library.   The *ANU PQ*  package, written by E.  O'Brien, consists of a C
    program implementing  a <p>-quotient and a <p>-group generation algorithm
    and functions to interface  this program with GAP (or  Cayley).  The *NQ*
    package, written by W.  Nickel, consists of  a C  program implementing an
    algorithm  to  compute  the  largest  nilpotent quotient  of  a  finitely
    presented group and a function to call this program from GAP.  The *Weyl*
    package, written  by M.  Geck, contains functions to compute with  finite
    Weyl   groups,   associated   (Iwahori-)  Hecke   algebras,   and   their
    representations.


What's New in 3.2
=================

    It  is  now possible to  extract  several  elements  from  a list  with a
    construct similar to  the one used to extract single elements.  This also
    works recursively,  so  that  it is  for  example possible  to extract  a
    submatrix of a matrix.  It is also possible to assign several elements to
    a list at once.

    Permutations can now operate on more than 65536 points.

    Ranges can now also have increments other than 1, i.e., a range is  now a
    dense  list  of  integers  such  that  the  difference  between  any  two
    consecutive elements is a nonzero constant.

    Strings are now also lists, namely  lists of characters, which are  a new
    builtin  datatype.   This  makes  functions  easier  to  write that  deal
    extensively with strings, such as 'DisplayCharTable'.

    GAP  now  supports  *univariate  polynomials*  over arbitrary coefficient
    rings.  Since the coefficient  ring may itself be a polynomial ring it is
    possible to create multivariate polynomial rings, though this is not very
    efficient.  Polynomials are implemented  in the GAP programming language,
    but there are supporting kernel functions to improve efficiency.

    Previously  the  entries  of  a  matrix  had  to be  among  the  built-in
    datatypes, i.e., rationals, cyclotomics, and finite field elements.  This
    restriction  has been removed, so that it is  now possible for example to
    compute with matrices whose entries are polynomials.

    There is now an  implementation of the  Dixon-Schneider algorithm,  which
    computes the character table of an arbitrary group.

    For permutation  groups  there are new functions to test if a permutation
    group is solvable,  and  if so  to find  a power-commutator presentation.
    Also there  is a  new  function  to compute  the composition series of  a
    permutation group.

    The  functions   to  compute  presentations  for  subgroups  of  finitely
    presented groups and to simplify them are new.

    There are new functions  that work with  table  of  marks, which  give  a
    compact description of the subgroup  lattice  of  a group.   For  example
    there is  a function that computes the  value of the Moebius function for
    the subgroup lattice of a group with a given table of marks.

    E. O'Brien and C. Rhodes provided  a library of 3-groups of size dividing
    729.   The character  table library  has  been extended  by  about 60 new
    ordinary tables and  about 200 new modular tables.  There is  also a data
    library that contains table of marks for various groups, e.g., McL.

    The share library packages  *ANU PQ*,  *NQ*, and  *Weyl* mentioned in the
    previous section are also new.


How to get GAP
==============
                                     Ceterum censeo:
                                       Nobody has ever paid a licence fee
                                         for using a proof
                                       that shows Sylow's subgroups to exist.
                                       Nobody should ever pay a licence fee
                                         for using a program
                                       that computes Sylow's subgroups.
                                                               (J. Neub"user)

    GAP  is distributed *free  of  charge*.  You  can obtain it via  'ftp' or
    electronic mail and give it away to your colleagues.  GAP is *not* in the
    public domain, however.  In particular you are not allowed to incorporate
    GAP or parts thereof into a commercial product.

    If you get GAP, we would appreciate it  if you could notify us,  e.g., by
    sending  a  short  e-mail  message  to  'gap@samson.math.rwth-aachen.de',
    containing your  full  name and address,  so that we have a rough idea of
    the number of users.  We also hope that this number will be  large enough
    to convince various  agencies that GAP is a project worthy of (financial)
    support.   If you publish some result that was partly obtained using GAP,
    we would  appreciate  it  if you  would cite GAP,  just as you would cite
    another paper that you  used.  Again  we  would  appreciate if  you could
    inform us about such a paper.

    We  distribute the  *full  source*  for  everything, the  C code for  the
    kernel, the GAP code for the library, and the LaTeX  code for the manual,
    which has  at present about 800 pages.  So it should be no problem to get
    GAP,  even if you have a rather uncommon system.  Of course, ports to non
    UNIX systems may require some work.   We already  have ports  for  IBM PC
    compatibles with an Intel 80386 or 80486 and for  the Atari ST.   We also
    hope to provide a port  of  GAP 3.2 to the Apple  Macintosh  in the  near
    future (there is already  a port of GAP 3.1).  Note that about 4 MByte of
    main memory and a harddisk are required to run GAP.

    GAP 3.2 can be obtained by anonymous *ftp* from the following servers.

    'samson.math.rwth-aachen.de':
            Lehrstuhl D fur Mathematik, RWTH Aachen, Germany (137.226.152.6).

    'dimacs.rutgers.edu':
            DIMACS, Rutgers, New Brunswick, New Jersey (128.6.75.16).

    'math.ucla.edu':
            Math. Dept., Univ. of California at Los Angeles (128.97.4.254).

    'wuarchive.wustl.edu':
    	    Mathematics Archives, Univ. of Tennessee (128.252.135.4,
            directory '/edu/math/source.code/group.theory/gap').

    'pell.anu.edu.au':
            Math. Research Section, Australian National Univ. (150.203.15.5).

    'ftp' to the server  *closest* to  you, login as user 'ftp' and give your
    full  e-mail  address  as password.  GAP  is in the  directory 'pub/gap'.
    Remember when you transmit  the files  to set the  file  transfer type to
    *binary image*, otherwise you will only receive unusable  garbage.  Those
    servers will always have the latest version of GAP available.

    GAP can also be obtained via *electronic mail*.  To get  one of the files
    mentioned  below  send a message to 'listserv@samson.math.rwth-aachen.de'
    containing a line  'get GAP <file-name>',  e.g., 'get  GAP src3r2.tar.Z'.
    'listserv' will reply by sending you the file as e-mail message.

    Because most files are  large  binary files they will  be  uuencoded  and
    split into  several parts,  each   at  most 64  kBytes  large.   You  can
    concatenate  the parts  by hand,  removing the mail  header, and then use
    'uudecode' to decode them.  We suggest however that you also get 'uud.c',
    which skips  the  mail headers automatically  and  is also able to fix up
    transmission errors caused by 'EBCDIC' machines.  You can also get single
    parts of a file by sending 'get GAP <file-name> <part-nr>'.

    For users in the United Kingdom with only Janet access, neither 'ftp' nor
    the mail server will work (please do *not*  try to  use the mail server).
    Please contact Derek Holt (e-mail address 'dfh@maths.warwick.ac.uk').  He
    has kindly offered us to distribute GAP in the United Kingdom.

    The 'ftp'  directory and  the 'listserv' archive   contain  the following
    files.  Please check first  which files you  need, to  avoid transferring
    those that you don't need.

    'README':               the file you are currently reading.

    GAP version 3 release 2 itself  comes in several files.   You do not need
    all of those files.  All files are 'compress'-ed 'tar' archives.

    'src3r2.tar.Z':         the *source code* for the GAP  kernel.  You  need
                            this unless you get one of the executables below.
                            This file is about 750 KBytes long.

    'lib3r2.tar.Z':         the *library of functions*.  You need this.  This
                            file is about 1000 KBytes long.

    'doc3r2.tar.Z':         the  *documentation*.  Serves  as  LaTeX   source
                            for the printed manual and  online documentation.
                            Contains further installation  information.  This
                            file is about 850 KBytes long.

    'doc3r2.dvi.Z':         the preformatted  documentation.  You  need  this
                            if you do not have a  *big*  TeX.  This  file  is
                            about 1100 KByte long.

    'grp3r2.tar.Z':         various *group libraries*.  Contains for  example
                            all primitive permutation  groups  of  degree  at
                            most 50.  This file is about 50 KByte long.

    'two3r2.tar.Z':         the library of *2-group* of  size  at  most  256.
                            This file is about 650 KByte long.

    'thr3r2.tar.Z':         the library of *3-groups* of  size at  most  729.
                            This file is about 20 KByte long.

    'tbl3r2.tar.Z':         a library of *character tables* including all  of
                            the ATLAS.  This file is about 2050 KByte long.

    'tom3r2.tar.Z':         a library of *table of marks* of various  groups.
                            This file is about 450 KByte long.

    'anupq.tar.Z':          the *ANU PQ* share library package.  This file is
                            about 350 KByte long.

    'nq.tar.Z':             the *NQ*  share  library  package.  This  file is
                            about 100 KByte long.

    'weyl.tar.Z':           the *Weyl* share library package.  This  file  is
                            about 50 KByte long.

    'src3r2.zoo', 'lib3r2.zoo', 'doc3r2.zoo', 'grp3r2.zoo'
    'tbl3r2.zoo', 'two3r2.zoo', 'thr3r2.zoo', 'tom3r2.zoo',
    'anupq.zoo',  'nq.zoo',     'weyl.zoo':
                            'zoo'  archives  containing  *exactly*  the  same
                            files as the 'compress'-ed 'tar' archives  above.
                            The advantage of 'compress'-ed 'tar'  archives is
                            that  'uncompress' and 'tar' are widely available
                            on UNIX systems.  The advantage of 'zoo' archives
                            is  that they are smaller (about  30 percent) and
                            that 'zoo' is more common on PC-s and Atari ST-s.
                            (These files may not be available on all servers)

    We  supply  executables  for  machines  that don't usually  come with a C
    compiler or machines where  the  standard  C  compiler does  not  produce
    optimal results.  If you have one of those machines it will be easier for
    you  to  get  this executable  instead  of compiling  GAP  yourself.  The
    following  executables  are  available  (again  these  files  may not  be
    available on all servers)

    'gapexe.386':           executable for IBM PC compatibles with  an  Intel
                            80386 or  80486 running MS-DOS 5.0 compiled  with
                            the GNU  C  2.2.2  compiler.   See below  for the
                            copyright.  This file is about 500 KByte long.

    'gapexe.next':          executable  for the NeXT (680?0) running NeXTstep
                            3.0  compiled  with  GNU C 2.3.3 compiler.   This
                            file is about 400 KByte long.

    'gapexe.st':            executable  for  Atari  ST  (680?0)  running  TOS
                            compiled with the GNU C compiler.  This  file  is
                            about 450 KByte long.

    'gapexe.su3':           executable for SUN 3 (680?0) running SunOS 4.0 or
                            higher compiled  with the  GNU C  compiler.  This
                            file is about 500 KByte long.

    'gapexe.su4':           executable for SUN 4 (Sparc) running SunOS 4.1 or
                            higher  compiled with the GNU  C  2.3.2 compiler.
                            This file is about 600 KByte long.

    The following support files are also available (and again these files may
    not be available on all servers)

    'compress.tar':         'compress' version 4.1.  You  need  this  program
                            to uncompress  the  compressed  tar  files.  Note
                            however, that almost all UNIX systems  these days
                            already come with an executable 'compress'.  This
                            file is about 90 KByte long.

    'patch.tar.Z':          Larry  Wall's  'patch'  program  version  2.0.2.0
                            (patchlevel 12u4).  This program  can  be used to
                            automatically  apply upgrades.   Note  that older
                            versions  of 'patch' are *not* able to understand
                            the  unified 'diff'  format  used  in the upgrade
                            files.  This file is about 70 KByte long.

    'uud.c':                'uud' version 3.4.  'uud' is much better than the
                            'uudecode' that comes  with  most  UNIX  systems.
                            This file is about 12 KByte long.

    'zoo21.tar.Z':          Rahul Dhesi's  'zoo' archiver  version  2.1.  You
                            need this  to  unpack  the  *zoo-archives*.  Note
                            that the widespread version 2.01 will *not* work.
                            This file is about 250 KByte long.

    'zooexe.386':           Executable of 'zoo' for IBM PC compatibles.  This
                            file is about 55 KByte long.

    'zooexe.st':            Executable of 'zoo' for the Atari ST.  This  file
                            is about 80 KByte long.


How to install GAP
==================

    The file  'install.tex' in 'doc3r2.tar.Z' contains extensive installation
    instructions.  If however, you are one of those  who never  read manuals,
    here is a quick installation guide.

    First for UNIX.

    Make a directory for GAP,  e.g., '~/gap/'  or  '/usr/local/lib/gap/'.

    Unpack the source  archive  'src3r2.tar.Z' into the subdirectory  'src/';
    unpack the library archive  'lib3r2.tar.Z' into the subdirectory  'lib/';
    unpack the documentation    'doc3r2.tar.Z' into the subdirectory  'doc/'.

    If you  have obtained the optional groups and character tables  libraries
    'grp3r2.tar.Z',   'tbl3r2.tar.Z',  'two3r2.tar.Z',   'thr3r2.tar.Z',  and
    'tom3r2.tar.Z',  unpack  them  into the  subdirectories  'grp/',  'tbl/',
    'two/', 'thr/', or 'tom/'.

    Change into 'src/' and execute 'make' to  see a list of possible targets;
    select a target, if in doubt use 'bsd' or 'usg', and make the kernel.

    In an appropriate directory, e.g., '~/bin/' or  '/usr/local/bin/', create
    a shell script that executes the GAP kernel.  This should look like

       exec <gap-directory>/src/gap  -m 4m  -l <gap-directory>/lib/  $*

    The option '-m' specifies the amount  of initial memory; the  option '-l'
    specifies where to find the library, if you get it wrong GAP complains

       gap: hmm, I cannot find 'lib/init.g', maybe use option '-l <libname>'?

    Change into 'doc/' and make the printed manual with the commands

       latex manual;  latex manual;  lp -dvi manual.dvi

    or something similar, according to your local custom for using LaTeX.

    Try something in GAP,  e.g., the following exercises GAP quite a bit

       gap> m11 := Group( (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6) );;
       gap> Number( ConjugacyClasses( m11 ) );

    The result should be 10.

    Next for IBM PC compatibles with an Intel 80386 or 80486 running  MS-DOS.

    Make a directory for GAP, e.g, 'c:\gap\'.

    Put the executable 'gapexe.386' into this directory calling it 'gap.exe'.
    Unpack the library archive   'lib3r2.zoo'  into the subdirectory  'lib\';
    unpack the documentation     'doc3r2.zoo'  into the subdirectory  'doc\'.

    If  you have  obtained the optional  groups and character table libraries
    'grp3r2.zoo', 'tbl3r2.zoo', 'two3r2.zoo', 'thr3r2.zoo', and 'tom3r2.zoo',
    unpack  them into the subdirectories 'grp\', 'tbl\', 'two\',  'thr\', and
    'tom\'.

    In a directory in  your  path,  e.g.,  'c:\bin\',  create  a  batch  file
    'gap.bat' that executes the GAP kernel.  This should look like

       <gap-directory>\gap  -m 4m  -l <gap-directory>\lib\  %1 %2 %3 %4

    The option '-m' specifies the amount  of initial memory; the  option '-l'
    specifies where to find the library, if you get it wrong GAP complains

       gap: hmm, I cannot find 'lib/init.g', maybe use option '-l <libname>'?

    Add the following line to your 'autoexec.bat' file

       SET GO32TMP=<swap-file-directory>

    where <swap-file-directory> should be the directory where you want GAP to
    put the  swap  file,  e.g.,  'c:\tmp'.   The swap  file  will  be  called
    'page????.386' and  is normally removed when GAP exits.  If 'GO32TMP'  is
    not set, 'GCCTMP', 'TMP', 'TEMP' are checked (in this order).  If neither
    is  set, GAP  will not swap to  disk.  *Note  that you must reboot before
    this change in 'autoexec.bat' takes effect*.

    Change into 'doc\' and make the printed manual with the commands

       latex manual;  latex manual;  print manual.dvi

    or something similar, according to your local custom for using LaTeX.

    Try something in GAP,  e.g., the following exercises GAP quite a bit

       gap> m11 := Group( (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6) );;
       gap> Number( ConjugacyClasses( m11 ) );

    The result should be 10.

    Note that GAP for the 386  will  use up  to 128 MByte of extended  memory
    (using XMS, VDISK memory  allocation  strategies)  or up to 128  MByte of
    expanded memory (using VCPI programs, such as QEMM  and 386MAX) and up to
    128 MByte of disk space for swapping.   Further note that GAP for the 386
    will *not* run under Windows (because it does not support DPMI).

    If you hit <ctr>-'C' the DOS  extender ('go32') catches it and aborts GAP
    immediately.  The  keys  <ctr>-'Z' and  <alt>-'C'  can be used instead to
    interupt GAP.

    The arrow keys <left>, <right>, <up>, <down>, <home>, <end>, and <delete>
    can be used for command line editing with their intuitive meaning.

    Pathnames may be  given inside GAP using either shlash ('/') or backslash
    ('\') as a separator (though '\' must be escaped in strings of course).

    The system dependent  part of GAP for the 386 ('sysdos.c') was written by
    Steve Linton (111  Ross  St.,  Cambridge,  CB1 3BS, UK,  +44 223  411661,
    'sl25@cus.cam.ac.uk').  He assignes the copyright to the Lehrstuhl D fuer
    Mathematik.  Many thanks to Steve Linton for his work.

    GAP for the 386 was compiled  with DJ Delorie's port of the Free Software
    Foundation's GNU C compiler version 2.1.  The compiler can be obtained by
    anonymous  'ftp'  from  'grape.ecs.clarkson.edu'  where   it  is  in  the
    directory 'pub/msdos/djgpp'.  Many thanks to the Free Software Foundation
    and DJ Delorie for this amazing piece of work.

    The GNU C compiler is

        Copyright (C) 1989 Free Software Foundation, Inc.
                           675 Mass Ave, Cambridge, MA 02139, USA

    under the  terms  of the GNU General Public License (GPL).  Note that the
    GNU  GPL  states  that the  mere act  of compiling  does not  affect  the
    copyright status of GAP.

    The modifications to the compiler to make it operating under  MS-DOS, the
    functions from  the standard library 'libpc.a', the  modifications of the
    functions from the standard library  'libc.a' to make  them operate under
    MS-DOS, and the DOS extender  'go32' (which is prepended to 'gapexe.386')
    are

        Copyright (C) 1991 DJ Delorie,
                           24 Kirsten Ave, Rochester NH 03867-2954, USA

    also under  the terms of the GNU GPL.  The terms of  the GPL require that
    we make the source code for 'libpc.a' available.  They can be obtained by
    writing to Steve Linton (however, it may be  easier for you to 'ftp' them
    from  'grape.ecs.clarkson.edu'  yourself).   They  also require that  GAP
    falls under the GPL too, i.e., is distributed free   coof users.  We also hope that this number will be  large enough
    to convince various  agencies that GAP is a project worthy of (financial)
    support.   If you publish some result that was partly obtained using GAP,
    we would  appreciate  it  if you  would cite GAP,  just as you would cite
    another paper that you  used.  Again  we  would  appreciate if  you could
    inform us about such a paper.

    We  distribute the  *full  source*  for  everything, the  C code for  the
    kernel, the GAP code for the library, and the LaTeX  code for the manual,
    which has  at present about 800 pages.  So it should be no problem to get
    GAP,  even if you have a rather uncommon system.  Of course, ports to non
    UNIX systems may require some work.   We already  have ports  for  IBM PC
    compatibles with an Intel 80386 or 80486 and for  the Atari ST.   We also
    hope to provide a port  of  GAP 3.2 to the Apple  Macintosh  in the  near
    future (there is already  a port of GAP 3.1).  Note that about 4 MByte of
    main memory and a harddisk are required to run GAP.

    GAP 3.2 can be obtained by anonymous *ftp* from the following servers.

    'samson.math.rwth-aachen.de':
            Lehrstuhl D fur Mathematik, RWTH Aachen, Germany (137.226.152.6).

    'dimacs.rutgers.edu':
            DIMACS, Rutgers, New Brunswick, New Jersey (128.6.75.16).

    'math.ucla.edu':
            Math. Dept., Univ. of California at Los Angeles (128.97.4.254).

    'wuarchive.wustl.edu':
    	    Mathematics Archives, Univ. of Tennessee (128.252.135.4,
            directory '/edu/math/source.code/group.theory/gap').

    'pell.anu.edu.au':
            Math. Research Section, Australian National Univ. (150.203.15.5).

    'ftp' to the server  *closest* to  you, login as user 'ftp' and give your
    full  e-mail  address  as password.  GAP  is in the  directory 'pub/gap'.
    Remember when you transmit  the files  to set the  file  transfer type to
    *binary image*, otherwise you will only receive unusable  garbage.  Those
    servers will always have the latest version of GAP available.

    GAP can also be obtained via *electronic mail*.  To get  one of the files
    mentioned  below  send a message to 'listserv@samson.math.rwth-aachen.de'
    containing a line  'get GAP <file-name>',  e.g., 'get  GAP src3r2.tar.Z'.
    'listserv' will reply by sending you the file as e-mail message.

    Because most files are  large  binary files they will  be  uuencoded  and
    split into  several parts,  each   at  most 64  kBytes  large.   You  can
    concatenate  the parts  by hand,  removing the mail  header, and then use
    'uudecode' to decode them.  We suggest however that you also get 'uud.c',
    which skips  the  mail headers automatically  and  is also able to fix up
    transmission errors caused by 'EBCDIC' machines.  You can also get single
    parts of a file by sending 'get GAP <file-name> <part-nr>'.

    For users in the United Kingdom with only Janet access, neither 'ftp' nor
    the mail server will work (please do *not*  try to  use the mail server).
    Please contact Derek Holt (e-mail address 'dfh@maths.warwick.ac.uk').  He
    has kindly offered us to distribute GAP in the United Kingdom.

    The 'ftp'  directory and  the 'listserv' archive   contain  the following
    files.  Please check first  which files you  need, to  avoid transferring
    those that you don't need.

    'README':               the file you are currently reading.

    GAP version 3 release 2 itself  comes in several files.   You do not need
    all of those files.  All files are 'compress'-ed 'tar' archives.

    'src3r2.tar.Z':         the *source code* for the GAP  kernel.  You  need
                            this unless you get one of the executables below.
                            This file is about 750 KBytes long.

    'lib3r2.tar.Z':         the *library of functions*.  You need this.  This
                            file is about 1000 KBytes long.

    'doc3r2.tar.Z':         the  *documentation*.  Serves  as  LaTeX   source
                            for the printed manual and  online documentation.
                            Contains further installation  information.  This
                            file is about 850 KBytes long.

    'doc3r2.dvi.Z':         the preformatted  documentation.  You  need  this
                            if you do not have a  *big*  TeX.  This  file  is
                            about 1100 KByte long.

    'grp3r2.tar.Z':         various *group libraries*.  Contains for  example
                            all primitive permutation  groups  of  degree  at
                            most 50.  This file is about 50 KByte long.

    'two3r2.tar.Z':         the library of *2-group* of  size  at  most  256.
                            This file is about 650 KByte long.

    'thr3r2.tar.Z':         the library of *3-groups* of  size at  most  729.
                            This file is about 20 KByte long.

    'tbl3r2.tar.Z':         a library of *character tables* including all  of
                            the ATLAS.  This file is about 2050 KByte long.

    'tom3r2.tar.Z':         a library of *table of marks* of various  groups.
                            This file is about 450 KByte long.

    'anupq.tar.Z':          the *ANU PQ* share library package.  This file is
                            about 350 KByte long.

    'nq.tar.Z':             the *NQ*  share  library  package.  This  file is
                            about 100 KByte long.

    'weyl.tar.Z':           the *Weyl* share library package.  This  file  is
                            about 50 KByte long.

    'src3r2.zoo', 'lib3r2.zoo', 'doc3r2.zoo', 'grp3r2.zoo'
    'tbl3r2.zoo', 'two3r2.zoo', 'thr3r2.zoo', 'tom3r2.zoo',
    'anupq.zoo',  'nq.zoo',     'weyl.zoo':
                            'zoo'  archives  containing  *exactly*  the  same
                            files as the 'compress'-ed 'tar' archives  above.
                            The advantage of 'compress'-ed 'tar'  archives is
                            that  'uncompress' and 'tar' are widely available
                            on UNIX systems.  The advantage of 'zoo' archives
                            is  that they are smaller (about  30 percent) and
                            that 'zoo' is more common on PC-s and Atari ST-s.
                            (These files may not be available on all servers)

    We  supply  executables  for  machines  that don't usually  come with a C
    compiler or machines where  the  standard  C  compiler does  not  produce
    optimal results.  If you have one of those machines it will be easier for
    you  to  get  this executable  instead  of compiling  GAP  yourself.  The
    following  executables  are  available  (again  these  files  may not  be
    available on all servers)

    'gapexe.386':           executable for IBM PC compatibles with  an  Intel
                            80386 or  80486 running MS-DOS 5.0 compiled  with
                            the GNU  C  2.2.2  compiler.   See below  for the
                            copyright.  This file is about 500 KByte long.

    'gapexe.next':          executable  for the NeXT (680?0) running NeXTstep
                            3.0  compiled  with  GNU C 2.3.3 compiler.   This
                            file is about 400 KByte long.

    'gapexe.st':            executable  for  Atari  ST  (680?0)  running  TOS
                            compiled with the GNU C compiler.  This  file  is
                            about 450 KByte long.

    'gapexe.su3':           executable for SUN 3 (680?0) running SunOS 4.0 or
                            higher compiled  with the  GNU C  compiler.  This
                            file is about 500 KByte long.

    'gapexe.su4':           executable for SUN 4 (Sparc) running SunOS 4.1 or
                            higher  compiled with the GNU  C  2.3.2 compiler.
                            This file is about 600 KByte long.

    The following support files are also available (and again these files may
    not be available on all servers)

    'compress.tar':         'compress' version 4.1.  You  need  this  program
                            to uncompress  the  compressed  tar  files.  Note
                            however, that almost all UNIX systems  these days
                            already come with an executable 'compress'.  This
                            file is about 90 KByte long.

    'patch.tar.Z':          Larry  Wall's  'patch'  program  version  2.0.2.0
                            (patchlevel 12u4).  This program  can  be used to
                            automatically  apply upgrades.   Note  that older
                            versions  of 'patch' are *not* able to understand
                            the  unified 'diff'  format  used  in the upgrade
                            files.  This file is about 70 KByte long.

    'uud.c':                'uud' version 3.4.  'uud' is much better than the
                            'uudecode' that comes  with  most  UNIX  systems.
                            This file is about 12 KByte long.

    'zoo21.tar.Z':          Rahul Dhesi's  'zoo' archiver  version  2.1.  You
                            need this  to  unpack  the  *zoo-archives*.  Note
                            that the widespread version 2.01 will *not* work.
                            This file is about 250 KByte long.

    'zooexe.386':           Executable of 'zoo' for IBM PC compatibles.  This
                            file is about 55 KByte long.

    'zooexe.st':            Executable of 'zoo' for the Atari ST.  This  file
                            is about 80 KByte long.


How to install GAP
==================

    The file  'install.tex' in 'doc3r2.tar.Z' contains extensive installation
    instructions.  If however, you are one of those  who never  read manuals,
    here is a quick installation guide.

    First for UNIX.

    Make a directory for GAP,  e.g., '~/gap/'  or  '/usr/local/lib/gap/'.

    Unpack the source  archive  'src3r2.tar.Z' into the subdirectory  'src/';
    unpack the library archive  'lib3r2.tar.Z' into the subdirectory  'lib/';
    unpack the documentation    'doc3r2.tar.Z' into the subdirectory  'doc/'.

    If you  have obtained the optional groups and character tables  libraries
    'grp3r2.tar.Z',   'tbl3r2.tar.Z',  'two3r2.tar.Z',   'thr3r2.tar.Z',  and
    'tom3r2.tar.Z',  unpack  them  into the  subdirectories  'grp/',  'tbl/',
    'two/', 'thr/', or 'tom/'.

    Change into 'src/' and execute 'make' to  see a list of possible targets;
    select a target, if in doubt use 'bsd' or 'usg', and make the kernel.

    In an appropriate directory, e.g., '~/bin/' or  '/usr/local/bin/', create
    a shell script that executes the GAP kernel.  This should look like

       exec <gap-directory>/src/gap  -m 4m  -l <gap-directory>/lib/  $*

    The option '-m' specifies the amount  of initial memory; the  option '-l'
    specifies where to find the library, if you get it wrong GAP complains

       gap: hmm, I cannot find 'lib/init.g', maybe use option '-l <libname>'?

    Change into 'doc/' and make the printed manual with the commands

       latex manual;  latex manual;  lp -dvi manual.dvi

    or something similar, according to your local custom for using LaTeX.

    Try something in GAP,  e.g., the following exercises GAP quite a bit

       gap> m11 := Group( (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6) );;
       gap> Number( ConjugacyClasses( m11 ) );

    The result should be 10.

    Next for IBM PC compatibles with an Intel 80386 or 80486 running  MS-DOS.

    Make a directory for GAP, e.g, 'c:\gap\'.

    Put the executable 'gapexe.386' into this directory calling it 'gap.exe'.
    Unpack the library archive   'lib3r2.zoo'  into the subdirectory  'lib\';
    unpack the documentation     'doc3r2.zoo'  into the subdirectory  'doc\'.

    If  you have  obtained the optional  groups and character table libraries
    'grp3r2.zoo', 'tbl3r2.zoo', 'two3r2.zoo', 'thr3r2.zoo', and 'tom3r2.zoo',
    unpack  them into the subdirectories 'grp\', 'tbl\', 'two\',  'thr\', and
    'tom\'.

    In a directory in  your  path,  e.g.,  'c:\bin\',  create  a  batch  file
    'gap.bat' that executes the GAP kernel.  This should look like

       <gap-directory>\gap  -m 4m  -l <gap-directory>\lib\  %1 %2 %3 %4

    The option '-m' specifies the amount  of initial memory; the  option '-l'
    specifies where to find the library, if you get it wrong GAP complains

       gap: hmm, I cannot find 'lib/init.g', maybe use option '-l <libname>'?

    Add the following line to your 'autoexec.bat' file

       SET GO32TMP=<swap-file-directory>

    where <swap-file-directory> should be the directory where you want GAP to
    put the  swap  file,  e.g.,  'c:\tmp'.   The swap  file  will  be  called
    'page????.386' and  is normally removed when GAP exits.  If 'GO32TMP'  is
    not set, 'GCCTMP', 'TMP', 'TEMP' are checked (in this order).  If neither
    is  set, GAP  will not swap to  disk.  *Note  that you must reboot before
    this change in 'autoexec.bat' takes effect*.

    Change into 'doc\' and make the printed manual with the commands

       latex manual;  latex manual;  print manual.dvi

    or something similar, according to your local custom for using LaTeX.

    Try something in GAP,  e.g., the following exercises GAP quite a bit

       gap> m11 := Group( (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6) );;
       gap> Number( ConjugacyClasses( m11 ) );

    The result should be 10.

    Note that GAP for the 386  will  use up  to 128 MByte of extended  memory
    (using XMS, VDISK memory  allocation  strategies)  or up to 128  MByte of
    expanded memory (using VCPI programs, such as QEMM  and 386MAX) and up to
    128 MByte of disk space for swapping.   Further note that GAP for the 386
    will *not* run under Windows (because it does not support DPMI).

    If you hit <ctr>-'C' the DOS  extender ('go32') catches it and aborts GAP
    immediately.  The  keys  <ctr>-'Z' and  <alt>-'C'  can be used instead to
    interupt GAP.

    The arrow keys <left>, <right>, <up>, <down>, <home>, <end>, and <delete>
    can be used for command line editing with their intuitive meaning.

    Pathnames may be  given inside GAP using either shlash ('/') or backslash
    ('\') as a separator (though '\' must be escaped in strings of course).

    The system dependent  part of GAP for the 386 ('sysdos.c') was written by
    Steve Linton (111  Ross  St.,  Cambridge,  CB1 3BS, UK,  +44 223  411661,
    'sl25@cus.cam.ac.uk').  He assignes the copyright to the Lehrstuhl D fuer
    Mathematik.  Many thanks to Steve Linton for his work.

    GAP for the 386 was compiled  with DJ Delorie's port of the Free Software
    Foundation's GNU C compiler version 2.1.  The compiler can be obtained by
    anonymous  'ftp'  from  'grape.ecs.clarkson.edu'  where   it  is  in  the
    directory 'pub/msdos/djgpp'.  Many thanks to the Free Software Foundation
    and DJ Delorie for this amazing piece of work.

    The GNU C compiler is

        Copyright (C) 1989 Free Software Foundation, Inc.
                           675 Mass Ave, Cambridge, MA 02139, USA

    under the  terms  of the GNU General Public License (GPL).  Note that the
    GNU  GPL  states  that the  mere act  of compiling  does not  affect  the
    copyright status of GAP.

    The modifications to the compiler to make it operating under  MS-DOS, the
    functions from  the standard library 'libpc.a', the  modifications of the
    functions from the standard library  'libc.a' to make  them operate under
    MS-DOS, and the DOS extender  'go32' (which is prepended to 'gapexe.386')
    are

        Copyright (C) 1991 DJ Delorie,
                           24 Kirsten Ave, Rochester NH 03867-2954, USA

    also under  the terms of the GNU GPL.  The terms of  the GPL require that
    we make the source code for 'libpc.a' available.  They can be obtained by
    writing to Steve Linton (however, it may be  easier for you to 'ftp' them
    from  'grape.ecs.clarkson.edu'  yourself).   They  also require that  GAP
    falls under the GPL too, i.e., is distributed free   coof users.  We also hope that this number will be  large enough
    to convince various  agencies that GAP is a project worthy of (financial)
    support.   If you publish some result that was partly obtained using GAP,
    we would  appreciate  it  if you  would cite GAP,  just as you would cite
    another paper that you  used.  Again  we  would  appreciate if  you could
    inform us about such a paper.

    We  distribute the  *full  source*  for  everything, the  C code for  the
    kernel, the GAP code for the library, and the LaTeX  code for the manual,
    which has  at present about 800 pages.  So it should be no problem to get
    GAP,  even if you have a rather uncommon system.  Of course, ports to non
    UNIX systems may require some work.   We already  have ports  for  IBM PC
    compatibles with an Intel 80386 or 80486 and for  the Atari ST.   We also
    hope to provide a port  of  GAP 3.2 to the Apple  Macintosh  in the  near
    future (there is already  a port of GAP 3.1).  Note that about 4 MByte of
    main memory and a harddisk are required to run GAP.

    GAP 3.2 can be obtained by anonymous *ftp* from the following servers.

    'samson.math.rwth-aachen.de':
            Lehrstuhl D fur Mathematik, RWTH Aachen, Germany (137.226.152.6).

    'dimacs.rutgers.edu':
            DIMACS, Rutgers, New Brunswick, New Jersey (128.6.75.16).

    'math.ucla.edu':
            Math. Dept., Univ. of California at Los Angeles (128.97.4.254).

    'wuarchive.wustl.edu':
    	    Mathematics Archives, Univ. of Tennessee (128.252.135.4,
            directory '/edu/math/source.code/group.theory/gap').

    'pell.anu.edu.au':
            Math. Research Section, Australian National Univ. (150.203.15.5).

    'ftp' to the server  *closest* to  you, login as user 'ftp' and give your
    full  e-mail  address  as password.  GAP  is in the  directory 'pub/gap'.
    Remember when you transmit  the files  to set the  file  transfer type to
    *binary image*, otherwise you will only receive unusable  garbage.  Those
    servers will always have the latest version of GAP available.

    GAP can also be obtained via *electronic mail*.  To get  one of the files
    mentioned  below  send a message to 'listserv@samson.math.rwth-aachen.de'
    containing a line  'get GAP <file-name>',  e.g., 'get  GAP src3r2.tar.Z'.
    'listserv' will reply by sending you the file as e-mail message.

    Because most files are  large  binary files they will  be  uuencoded  and
    split into  several parts,  each   at  most 64  kBytes  large.   You  can
    concatenate  the parts  by hand,  removing the mail  header, and then use
    'uudecode' to decode them.  We suggest however that you also get 'uud.c',
    which skips  the  mail headers automatically  and  is also able to fix up
    transmission errors caused by 'EBCDIC' machines.  You can also get single
    parts of a file by sending 'get GAP <file-name> <part-nr>'.

    For users in the United Kingdom with only Janet access, neither 'ftp' nor
    the mail server will work (please do *not*  try to  use the mail server).
    Please contact Derek Holt (e-mail address 'dfh@maths.warwick.ac.uk').  He
    has kindly offered us to distribute GAP in the United Kingdom.

    The 'ftp'  directory and  the 'listserv' archive   contain  the following
    files.  Please check first  which files you  need, to  avoid transferring
    those that you don't need.

    'README':               the file you are currently reading.

    GAP version 3 release 2 itself  comes in several files.   You do not need
    all of those files.  All files are 'compress'-ed 'tar' archives.

    'src3r2.tar.Z':         the *source code* for the GAP  kernel.  You  need
                            this unless you get one of the executables below.
                            This file is about 750 KBytes long.

    'lib3r2.tar.Z':         the *library of functions*.  You need this.  This
                            file is about 1000 KBytes long.

    'doc3r2.tar.Z':         the  *documentation*.  Serves  as  LaTeX   source
                            for the printed manual and  online documentation.
                            Contains further installation  information.  This
                            file is about 850 KBytes long.

    'doc3r2.dvi.Z':         the preformatted  documentation.  You  need  this
                            if you do not have a  *big*  TeX.  This  file  is
                            about 1100 KByte long.

    'grp3r2.tar.Z':         various *group libraries*.  Contains for  example
                            all primitive permutation  groups  of  degree  at
                            most 50.  This file is about 50 KByte long.

    'two3r2.tar.Z':         the library of *2-group* of  size  at  most  256.
                            This file is about 650 KByte long.

    'thr3r2.tar.Z':         the library of *3-groups* of  size at  most  729.
                            This file is about 20 KByte long.

    'tbl3r2.tar.Z':         a library of *character tables* including all  of
                            the ATLAS.  This file is about 2050 KByte long.

    'tom3r2.tar.Z':         a library of *table of marks* of various  groups.
                            This file is about 450 KByte long.

    'anupq.tar.Z':          the *ANU PQ* share library package.  This file is
                            about 350 KByte long.

    'nq.tar.Z':             the *NQ*  share  library  package.  This  file is
                            about 100 KByte long.

    'weyl.tar.Z':           the *Weyl* share library package.  This  file  is
                            about 50 KByte long.

    'src3r2.zoo', 'lib3r2.zoo', 'doc3r2.zoo', 'grp3r2.zoo'
    'tbl3r2.zoo', 'two3r2.zoo', 'thr3r2.zoo', 'tom3r2.zoo',
    'anupq.zoo',  'nq.zoo',     'weyl.zoo':
                            'zoo'  archives  containing  *exactly*  the  same
                            files as the 'compress'-ed 'tar' archives  above.
                            The advantage of 'compress'-ed 'tar'  archives is
                            that  'uncompress' and 'tar' are widely available
                            on UNIX systems.  The advantage of 'zoo' archives
                            is  that they are smaller (about  30 percent) and
                            that 'zoo' is more common on PC-s and Atari ST-s.
                            (These files may not be available on all servers)

    We  supply  executables  for  machines  that don't usually  come with a C
    compiler or machines where  the  standard  C  compiler does  not  produce
    optimal results.  If you have one of those machines it will be easier for
    you  to  get  this executable  instead  of compiling  GAP  yourself.  The
    following  executables  are  available  (again  these  files  may not  be
    available on all servers)

    'gapexe.386':           executable for IBM PC compatibles with  an  Intel
                            80386 or  80486 running MS-DOS 5.0 compiled  with
                            the GNU  C  2.2.2  compiler.   See below  for the
                            copyright.  This file is about 500 KByte long.

    'gapexe.next':          executable  for the NeXT (680?0) running NeXTstep
                            3.0  compiled  with  GNU C 2.3.3 compiler.   This
                            file is about 400 KByte long.

    'gapexe.st':            executable  for  Atari  ST  (680?0)  running  TOS
                            compiled with the GNU C compiler.  This  file  is
                            about 450 KByte long.

    'gapexe.su3':           executable for SUN 3 (680?0) running SunOS 4.0 or
                            higher compiled  with the  GNU C  compiler.  This
                            file is about 500 KByte long.

    'gapexe.su4':           executable for SUN 4 (Sparc) running SunOS 4.1 or
                            higher  compiled with the GNU  C  2.3.2 compiler.
                            This file is about 600 KByte long.

    The following support files are also available (and again these files may
    not be available on all servers)

    'compress.tar':         'compress' version 4.1.  You  need  this  program
                            to uncompress  the  compressed  tar  files.  Note
                            however, that almost all UNIX systems  these days
                            already come with an executable 'compress'.  This
                            file is about 90 KByte long.

    'patch.tar.Z':          Larry  Wall's  'patch'  program  version  2.0.2.0
                            (patchlevel 12u4).  This program  can  be used to
                            automatically  apply upgrades.   Note  that older
                            versions  of 'patch' are *not* able to understand
                            the  unified 'diff'  format  used  in the upgrade
                            files.  This file is about 70 KByte long.

    'uud.c':                'uud' version 3.4.  'uud' is much better than the
                            'uudecode' that comes  with  most  UNIX  systems.
                            This file is about 12 KByte long.

    'zoo21.tar.Z':          Rahul Dhesi's  'zoo' archiver  version  2.1.  You
                            need this  to  unpack  the  *zoo-archives*.  Note
                            that the widespread version 2.01 will *not* work.
                            This file is about 250 KByte long.

    'zooexe.386':           Executable of 'zoo' for IBM PC compatibles.  This
                            file is about 55 KByte long.

    'zooexe.st':            Executable of 'zoo' for the Atari ST.  This  file
                            is about 80 KByte long.


How to install GAP
==================

    The file  'install.tex' in 'doc3r2.tar.Z' contains extensive installation
    instructions.  If however, you are one of those  who never  read manuals,
    here is a quick installation guide.

    First for UNIX.

    Make a directory for GAP,  e.g., '~/gap/'  or  '/usr/local/lib/gap/'.

    Unpack the source  archive  'src3r2.tar.Z' into the subdirectory  'src/';
    unpack the library archive  'lib3r2.tar.Z' into the subdirectory  'lib/';
    unpack the documentation    'doc3r2.tar.Z' into the subdirectory  'doc/'.

    If you  have obtained the optional groups and character tables  libraries
    'grp3r2.tar.Z',   'tbl3r2.tar.Z',  'two3r2.tar.Z',   'thr3r2.tar.Z',  and
    'tom3r2.tar.Z',  unpack  them  into the  subdirectories  'grp/',  'tbl/',
    'two/', 'thr/', or 'tom/'.

    Change into 'src/' and execute 'make' to  see a list of possible targets;
    select a target, if in doubt use 'bsd' or 'usg', and make the kernel.

    In an appropriate directory, e.g., '~/bin/' or  '/usr/local/bin/', create
    a shell script that executes the GAP kernel.  This should look like

       exec <gap-directory>/src/gap  -m 4m  -l <gap-directory>/lib/  $*

    The option '-m' specifies the amount  of initial memory; the  option '-l'
    specifies where to find the library, if you get it wrong GAP complains

       gap: hmm, I cannot find 'lib/init.g', maybe use option '-l <libname>'?

    Change into 'doc/' and make the printed manual with the commands

       latex manual;  latex manual;  lp -dvi manual.dvi

    or something similar, according to your local custom for using LaTeX.

    Try something in GAP,  e.g., the following exercises GAP quite a bit

       gap> m11 := Group( (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6) );;
       gap> Number( ConjugacyClasses( m11 ) );

    The result should be 10.

    Next for IBM PC compatibles with an Intel 80386 or 80486 running  MS-DOS.

    Make a directory for GAP, e.g, 'c:\gap\'.

    Put the executable 'gapexe.386' into this directory calling it 'gap.exe'.
    Unpack the library archive   'lib3r2.zoo'  into the subdirectory  'lib\';
    unpack the documentation     'doc3r2.zoo'  into the subdirectory  'doc\'.

    If  you have  obtained the optional  groups and character table libraries
    'grp3r2.zoo', 'tbl3r2.zoo', 'two3r2.zoo', 'thr3r2.zoo', and 'tom3r2.zoo',
    unpack  them into the subdirectories 'grp\', 'tbl\', 'two\',  'thr\', and
    'tom\'.

    In a directory in  your  path,  e.g.,  'c:\bin\',  create  a  batch  file
    'gap.bat' that executes the GAP kernel.  This should look like

       <gap-directory>\gap  -m 4m  -l <gap-directory>\lib\  %1 %2 %3 %4

    The option '-m' specifies the amount  of initial memory; the  option '-l'
    specifies where to find the library, if you get it wrong GAP complains

       gap: hmm, I cannot find 'lib/init.g', maybe use option '-l <libname>'?

    Add the following line to your 'autoexec.bat' file

       SET GO32TMP=<swap-file-directory>

    where <swap-file-directory> should be the directory where you want GAP to
    put the  swap  file,  e.g.,  'c:\tmp'.   The swap  file  will  be  called
    'page????.386' and  is normally removed when GAP exits.  If 'GO32TMP'  is
    not set, 'GCCTMP', 'TMP', 'TEMP' are checked (in this order).  If neither
    is  set, GAP  will not swap to  disk.  *Note  that you must reboot before
    this change in 'autoexec.bat' takes effect*.

    Change into 'doc\' and make the printed manual with the commands

       latex manual;  latex manual;  print manual.dvi

    or something similar, according to your local custom for using LaTeX.

    Try something in GAP,  e.g., the following exercises GAP quite a bit

       gap> m11 := Group( (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6) );;
       gap> Number( ConjugacyClasses( m11 ) );

    The result should be 10.

    Note that GAP for the 386  will  use up  to 128 MByte of extended  memory
    (using XMS, VDISK memory  allocation  strategies)  or up to 128  MByte of
    expanded memory (using VCPI programs, such as QEMM  and 386MAX) and up to
    128 MByte of disk space for swapping.   Further note that GAP for the 386
    will *not* run under Windows (because it does not support DPMI).

    If you hit <ctr>-'C' the DOS  extender ('go32') catches it and aborts GAP
    immediately.  The  keys  <ctr>-'Z' and  <alt>-'C'  can be used instead to
    interupt GAP.

    The arrow keys <left>, <right>, <up>, <down>, <home>, <end>, and <delete>
    can be used for command line editing with their intuitive meaning.

    Pathnames may be  given inside GAP using either shlash ('/') or backslash
    ('\') as a separator (though '\' must be escaped in strings of course).

    The system dependent  part of GAP for the 386 ('sysdos.c') was written by
    Steve Linton (111  Ross  St.,  Cambridge,  CB1 3BS, UK,  +44 223  411661,
    'sl25@cus.cam.ac.uk').  He assignes the copyright to the Lehrstuhl D fuer
    Mathematik.  Many thanks to Steve Linton for his work.

    GAP for the 386 was compiled  with DJ Delorie's port of the Free Software
    Foundation's GNU C compiler version 2.1.  The compiler can be obtained by
    anonymous  'ftp'  from  'grape.ecs.clarkson.edu'  where   it  is  in  the
    directory 'pub/msdos/djgpp'.  Many thanks to the Free Software Foundation
    and DJ Delorie for this amazing piece of work.

    The GNU C compiler is

        Copyright (C) 1989 Free Software Foundation, Inc.
                           675 Mass Ave, Cambridge, MA 02139, USA

    under the  terms  of the GNU General Public License (GPL).  Note that the
    GNU  GPL  states  that the  mere act  of compiling  does not  affect  the
    copyright status of GAP.

    The modifications to the compiler to make it operating under  MS-DOS, the
    functions from  the standard library 'libpc.a', the  modifications of the
    functions from the standard library  'libc.a' to make  them operate under
    MS-DOS, and the DOS extender  'go32' (which is prepended to 'gapexe.386')
    are

        Copyright (C) 1991 DJ Delorie,
                           24 Kirsten Ave, Rochester NH 03867-2954, USA

    also under  the terms of the GNU GPL.  The terms of  the GPL require that
    we make the source code for 'libpc.a' available.  They can be obtained by
    writing to Steve Linton (however, it may be  easier for you to 'ftp' them
    from  'grape.ecs.clarkson.edu'  yourself).   They  also require that  GAP
    falls under the GPL too, i.e., is distributed free   coof users.  We also hope that this number will be  large enough
    to convince various  agencies that GAP is a project worthy of (financial)
    support.   If you publish some result that was partly obtained using GAP,
    we would  appreciate  it  if you  would cite GAP,  just as you would cite
    another paper that you  used.  Again  we  would  appreciate if  you could
    inform us about such a paper.

    We  distribute the  *full  source*  for  everything, the  C code for  the
    kernel, the GAP code for the library, and the LaTeX  code for the manual,
    which has  at present about 800 pages.  So it should be no problem to get
    GAP,  even if you have a rather uncommon system.  Of course, ports to non
    UNIX systems may require some work.   We already  have ports  for  IBM PC
    compatibles with an Intel 80386 or 80486 and for  the Atari ST.   We also
    hope to provide a port  of  GAP 3.2 to the Apple  Macintosh  in the  near
    future (there is already  a port of GAP 3.1).  Note that about 4 MByte of
    main memory and a harddisk are required to run GAP.

    GAP 3.2 can be obtained by anonymous *ftp* from the following servers.

    'samson.math.rwth-aachen.de':
            Lehrstuhl D fur Mathematik, RWTH Aachen, Germany (137.226.152.6).

    'dimacs.rutgers.edu':
            DIMACS, Rutgers, New Brunswick, New Jersey (128.6.75.16).

    'math.ucla.edu':
            Math. Dept., Univ. of California at Los Angeles (128.97.4.254).

    'wuarchive.wustl.edu':
    	    Mathematics Archives, Univ. of Tennessee (128.252.135.4,
            directory '/edu/math/source.code/group.theory/gap').

    'pell.anu.edu.au':
            Math. Research Section, Australian National Univ. (150.203.15.5).

    'ftp' to the server  *closest* to  you, login as user 'ftp' and give your
    full  e-mail  address  as password.  GAP  is in the  directory 'pub/gap'.
    Remember when you transmit  the files  to set the  file  transfer type to
    *binary image*, otherwise you will only receive unusable  garbage.  Those
    servers will always have the latest version of GAP available.

    GAP can also be obtained via *electronic mail*.  To get  one of the files
    mentioned  below  send a message to 'listserv@samson.math.rwth-aachen.de'
    containing a line  'get GAP <file-name>',  e.g., 'get  GAP src3r2.tar.Z'.
    'listserv' will reply by sending you the file as e-mail message.

    Because most files are  large  binary files they will  be  uuencoded  and
    split into  several parts,  each   at  most 64  kBytes  large.   You  can
    concatenate  the parts  by hand,  removing the mail  header, and then use
    'uudecode' to decode them.  We suggest however that you also get 'uud.c',
    which skips  the  mail headers automatically  and  is also able to fix up
    transmission errors caused by 'EBCDIC' machines.  You can also get single
    parts of a file by sending 'get GAP <file-name> <part-nr>'.

    For users in the United Kingdom with only Janet access, neither 'ftp' nor
    the mail server will work (please do *not*  try to  use the mail server).
    Please contact Derek Holt (e-mail address 'dfh@maths.warwick.ac.uk').  He
    has kindly offered us to distribute GAP in the United Kingdom.

    The 'ftp'  directory and  the 'listserv' archive   contain  the following
    files.  Please check first  which files you  need, to  avoid transferring
    those that you don't need.

    'README':               the file you are currently reading.

    GAP version 3 release 2 itself  comes in several files.   You do not need
    all of those files.  All files are 'compress'-ed 'tar' archives.

    'src3r2.tar.Z':         the *source code* for the GAP  kernel.  You  need
                            this unless you get one of the executables below.
                            This file is about 750 KBytes long.

    'lib3r2.tar.Z':         the *library of functions*.  You need this.  This
                            file is about 1000 KBytes long.

    'doc3r2.tar.Z':         the  *documentation*.  Serves  as  LaTeX   source
                            for the printed manual and  online documentation.
                            Contains further installation  information.  This
                            file is about 850 KBytes long.

    'doc3r2.dvi.Z':         the preformatted  documentation.  You  need  this
                            if you do not have a  *big*  TeX.  This  file  is
                            about 1100 KByte long.

    'grp3r2.tar.Z':         various *group libraries*.  Contains for  example
                            all primitive permutation  groups  of  degree  at
                            most 50.  This file is about 50 KByte long.

    'two3r2.tar.Z':         the library of *2-group* of  size  at  most  256.
                            This file is about 650 KByte long.

    'thr3r2.tar.Z':         the library of *3-groups* of  size at  most  729.
                            This file is about 20 KByte long.

    'tbl3r2.tar.Z':         a library of *character tables* including all  of
                            the ATLAS.  This file is about 2050 KByte long.

    'tom3r2.tar.Z':         a library of *table of marks* of various  groups.
                            This file is about 450 KByte long.

    'anupq.tar.Z':          the *ANU PQ* share library package.  This file is
                            about 350 KByte long.

    'nq.tar.Z':             the *NQ*  share  library  package.  This  file is
                            about 100 KByte long.

    'weyl.tar.Z':           the *Weyl* share library package.  This  file  is
                            about 50 KByte long.

    'src3r2.zoo', 'lib3r2.zoo', 'doc3r2.zoo', 'grp3r2.zoo'
    'tbl3r2.zoo', 'two3r2.zoo', 'thr3r2.zoo', 'tom3r2.zoo',
    'anupq.zoo',  'nq.zoo',     'weyl.zoo':
                            'zoo'  archives  containing  *exactly*  the  same
                            files as the 'compress'-ed 'tar' archives  above.
                            The advantage of 'compress'-ed 'tar'  archives is
                            that  'uncompress' and 'tar' are widely available
                            on UNIX systems.  The advantage of 'zoo' archives
                            is  that they are smaller (about  30 percent) and
                            that 'zoo' is more common on PC-s and Atari ST-s.
                            (These files may not be available on all servers)

    We  supply  executables  for  machines  that don't usually  come with a C
    compiler or machines where  the  standard  C  compiler does  not  produce
    optimal results.  If you have one of those machines it will be easier for
    you  to  get  this executable  instead  of compiling  GAP  yourself.  The
    following  executables  are  available  (again  these  files  may not  be
    available on all servers)

    'gapexe.386':           executable for IBM PC compatibles with  an  Intel
                            80386 or  80486 running MS-DOS 5.0 compiled  with
                            the GNU  C  2.2.2  compiler.   See below  for the
                            copyright.  This file is about 500 KByte long.

    'gapexe.next':          executable  for the NeXT (680?0) running NeXTstep
                            3.0  compiled  with  GNU C 2.3.3 compiler.   This
                            file is about 400 KByte long.

    'gapexe.st':            executable  for  Atari  ST  (680?0)  running  TOS
                            compiled with the GNU C compiler.  This  file  is
                            about 450 KByte long.

    'gapexe.su3':           executable for SUN 3 (680?0) running SunOS 4.0 or
                            higher compiled  with the  GNU C  compiler.  This
                            file is about 500 KByte long.

    'gapexe.su4':           executable for SUN 4 (Sparc) running SunOS 4.1 or
                            higher  compiled with the GNU  C  2.3.2 compiler.
                            This file is about 600 KByte long.

    The following support files are also available (and again these files may
    not be available on all servers)

    'compress.tar':         'compress' version 4.1.  You  need  this  program
                            to uncompress  the  compressed  tar  files.  Note
                            however, that almost all UNIX systems  these days
                            already come with an executable 'compress'.  This
                            file is about 90 KByte long.

    'patch.tar.Z':          Larry  Wall's  'patch'  program  version  2.0.2.0
                            (patchlevel 12u4).  This program  can  be used to
                            automatically  apply upgrades.   Note  that older
                            versions  of 'patch' are *not* able to understand
                            the  unified 'diff'  format  used  in the upgrade
                            files.  This file is about 70 KByte long.

    'uud.c':                'uud' version 3.4.  'uud' is much better than the
                            'uudecode' that comes  with  most  UNIX  systems.
                            This file is about 12 KByte long.

    'zoo21.tar.Z':          Rahul Dhesi's  'zoo' archiver  version  2.1.  You
                            need this  to  unpack  the  *zoo-archives*.  Note
                            that the widespread version 2.01 will *not* work.
                            This file is about 250 KByte long.

    'zooexe.386':           Executable of 'zoo' for IBM PC compatibles.  This
                            file is about 55 KByte long.

    'zooexe.st':            Executable of 'zoo' for the Atari ST.  This  file
                            is about 80 KByte long.


How to install GAP
==================

    The file  'install.tex' in 'doc3r2.tar.Z' contains extensive installation
    instructions.  If however, you are one of those  who never  read manuals,
    here is a quick installation guide.

    First for UNIX.

    Make a directory for GAP,  e.g., '~/gap/'  or  '/usr/local/lib/gap/'.

    Unpack the source  archive  'src3r2.tar.Z' into the subdirectory  'src/';
    unpack the library archive  'lib3r2.tar.Z' into the subdirectory  'lib/';
    unpack the documentation    'doc3r2.tar.Z' into the subdirectory  'doc/'.

    If you  have obtained the optional groups and character tables  libraries
    'grp3r2.tar.Z',   'tbl3r2.tar.Z',  'two3r2.tar.Z',   'thr3r2.tar.Z',  and
    'tom3r2.tar.Z',  unpack  them  into the  subdirectories  'grp/',  'tbl/',
    'two/', 'thr/', or 'tom/'.

    Change into 'src/' and execute 'make' to  see a list of possible targets;
    select a target, if in doubt use 'bsd' or 'usg', and make the kernel.

    In an appropriate directory, e.g., '~/bin/' or  '/usr/local/bin/', create
    a shell script that executes the GAP kernel.  This should look like

       exec <gap-directory>/src/gap  -m 4m  -l <gap-directory>/lib/  $*

    The option '-m' specifies the amount  of initial memory; the  option '-l'
    specifies where to find the library, if you get it wrong GAP complains

       gap: hmm, I cannot find 'lib/init.g', maybe use option '-l <libname>'?

    Change into 'doc/' and make the printed manual with the commands

       latex manual;  latex manual;  lp -dvi manual.dvi

    or something similar, according to your local custom for using LaTeX.

    Try something in GAP,  e.g., the following exercises GAP quite a bit

       gap> m11 := Group( (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6) );;
       gap> Number( ConjugacyClasses( m11 ) );

    The result should be 10.

    Next for IBM PC compatibles with an Intel 80386 or 80486 running  MS-DOS.

    Make a directory for GAP, e.g, 'c:\gap\'.

    Put the executable 'gapexe.386' into this directory calling it 'gap.exe'.
    Unpack the library archive   'lib3r2.zoo'  into the subdirectory  'lib\';
    unpack the documentation     'doc3r2.zoo'  into the subdirectory  'doc\'.

    If  you have  obtained the optional  groups and character table libraries
    'grp3r2.zoo', 'tbl3r2.zoo', 'two3r2.zoo', 'thr3r2.zoo', and 'tom3r2.zoo',
    unpack  them into the subdirectories 'grp\', 'tbl\', 'two\',  'thr\', and
    'tom\'.

    In a directory in  your  path,  e.g.,  'c:\bin\',  create  a  batch  file
    'gap.bat' that executes the GAP kernel.  This should look like

       <gap-directory>\gap  -m 4m  -l <gap-directory>\lib\  %1 %2 %3 %4

    The option '-m' specifies the amount  of initial memory; the  option '-l'
    specifies where to find the library, if you get it wrong GAP complains

       gap: hmm, I cannot find 'lib/init.g', maybe use option '-l <libname>'?

    Add the following line to your 'autoexec.bat' file

       SET GO32TMP=<swap-file-directory>

    where <swap-file-directory> should be the directory where you want GAP to
    put the  swap  file,  e.g.,  'c:\tmp'.   The swap  file  will  be  called
    'page????.386' and  is normally removed when GAP exits.  If 'GO32TMP'  is
    not set, 'GCCTMP', 'TMP', 'TEMP' are checked (in this order).  If neither
    is  set, GAP  will not swap to  disk.  *Note  that you must reboot before
    this change in 'autoexec.bat' takes effect*.

    Change into 'doc\' and make the printed manual with the commands

       latex manual;  latex manual;  print manual.dvi

    or something similar, according to your local custom for using LaTeX.

    Try something in GAP,  e.g., the following exercises GAP quite a bit

       gap> m11 := Group( (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6) );;
       gap> Number( ConjugacyClasses( m11 ) );

    The result should be 10.

    Note that GAP for the 386  will  use up  to 128 MByte of extended  memory
    (using XMS, VDISK memory  allocation  strategies)  or up to 128  MByte of
    expanded memory (using VCPI programs, such as QEMM  and 386MAX) and up to
    128 MByte of disk space for swapping.   Further note that GAP for the 386
    will *not* run under Windows (because it does not support DPMI).

    If you hit <ctr>-'C' the DOS  extender ('go32') catches it and aborts GAP
    immediately.  The  keys  <ctr>-'Z' and  <alt>-'C'  can be used instead to
    interupt GAP.

    The arrow keys <left>, <right>, <up>, <down>, <home>, <end>, and <delete>
    can be used for command line editing with their intuitive meaning.

    Pathnames may be  given inside GAP using either shlash ('/') or backslash
    ('\') as a separator (though '\' must be escaped in strings of course).

    The system dependent  part of GAP for the 386 ('sysdos.c') was written by
    Steve Linton (111  Ross  St.,  Cambridge,  CB1 3BS, UK,  +44 223  411661,
    'sl25@cus.cam.ac.uk').  He assignes the copyright to the Lehrstuhl D fuer
    Mathematik.  Many thanks to Steve Linton for his work.

    GAP for the 386 was compiled  with DJ Delorie's port of the Free Software
    Foundation's GNU C compiler version 2.1.  The compiler can be obtained by
    anonymous  'ftp'  from  'grape.ecs.clarkson.edu'  where   it  is  in  the
    directory 'pub/msdos/djgpp'.  Many thanks to the Free Software Foundation
    and DJ Delorie for this amazing piece of work.

    The GNU C compiler is

        Copyright (C) 1989 Free Software Foundation, Inc.
                           675 Mass Ave, Cambridge, MA 02139, USA

    under the  terms  of the GNU General Public License (GPL).  Note that the
    GNU  GPL  states  that the  mere act  of compiling  does not  affect  the
    copyright status of GAP.

    The modifications to the compiler to make it operating under  MS-DOS, the
    functions from  the standard library 'libpc.a', the  modifications of the
    functions from the standard library  'libc.a' to make  them operate under
    MS-DOS, and the DOS extender  'go32' (which is prepended to 'gapexe.386')
    are

        Copyright (C) 1991 DJ Delorie,
                           24 Kirsten Ave, Rochester NH 03867-2954, USA

    also under  the terms of the GNU GPL.  The terms of  the GPL require that
    we make the source code for 'libpc.a' available.  They can be obtained by
    writing to Steve Linton (however, it may be  easier for you to 'ftp' them
    from  'grape.ecs.clarkson.edu'  yourself).   They  also require that  GAP
    falls under the GPL too, i.e., is distributed free   coof users.  We also hope that this number will be  large enough
    to convince various  agencies that GAP is a project worthy of (financial)
    support.   If you publish some result that was partly obtained using GAP,
    we would  appreciate  it  if you  would cite GAP,  just as you would cite
    another paper that you  used.  Again  we  would  appreciate if  you could
    inform us about such a paper.

    We  distribute the  *full  source*  for  everything, the  C code for  the
    kernel, the GAP code for the library, and the LaTeX  code for the manual,
    which has  at present about 800 pages.  So it should be no problem to get
    GAP,  even if you have a rather uncommon system.  Of course, ports to non
    UNIX systems may require some work.   We already  have ports  for  IBM PC
    compatibles with an Intel 80386 or 80486 and for  the Atari ST.   We also
    hope to provide a port  of  GAP 3.2 to the Apple  Macintosh  in the  near
    future (there is already  a port of GAP 3.1).  Note that about 4 MByte of
    main memory and a harddisk are required to run GAP.

    GAP 3.2 can be obtained by anonymous *ftp* from the following servers.

    'samson.math.rwth-aachen.de':
            Lehrstuhl D fur Mathematik, RWTH Aachen, Germany (137.226.152.6).

    'dimacs.rutgers.edu':
            DIMACS, Rutgers, New Brunswick, New Jersey (128.6.75.16).

    'math.ucla.edu':
            Math. Dept., Univ. of California at Los Angeles (128.97.4.254).

    'wuarchive.wustl.edu':
    	    Mathematics Archives, Univ. of Tennessee (128.252.135.4,
            directory '/edu/math/source.code/group.theory/gap').

    'pell.anu.edu.au':
            Math. Research Section, Australian National Univ. (150.203.15.5).

    'ftp' to the server  *closest* to  you, login as user 'ftp' and give your
    full  e-mail  address  as password.  GAP  is in the  directory 'pub/gap'.
    Remember when you transmit  the files  to set the  file  transfer type to
    *binary image*, otherwise you will only receive unusable  garbage.  Those
    servers will always have the latest version of GAP available.

    GAP can also be obtained via *electronic mail*.  To get  one of the files
    mentioned  below  send a message to 'listserv@samson.math.rwth-aachen.de'
    containing a line  'get GAP <file-name>',  e.g., 'get  GAP src3r2.tar.Z'.
    'listserv' will reply by sending you the file as e-mail message.

    Because most files are  large  binary files they will  be  uuencoded  and
    split into  several parts,  each   at  most 64  kBytes  large.   You  can
    concatenate  the parts  by hand,  removing the mail  header, and then use
    'uudecode' to decode them.  We suggest however that you also get 'uud.c',
    which skips  the  mail headers automatically  and  is also able to fix up
    transmission errors caused by 'EBCDIC' machines.  You can also get single
    parts of a file by sending 'get GAP <file-name> <part-nr>'.

    For users in the United Kingdom with only Janet access, neither 'ftp' nor
    the mail server will work (please do *not*  try to  use the mail server).
    Please contact Derek Holt (e-mail address 'dfh@maths.warwick.ac.uk').  He
    has kindly offered us to distribute GAP in the United Kingdom.

    The 'ftp'  directory and  the 'listserv' archive   contain  the following
    files.  Please check first  which files you  need, to  avoid transferring
    those that you don't need.

    'README':               the file you are currently reading.

    GAP version 3 release 2 itself  comes in several files.   You do not need
    all of those files.  All files are 'compress'-ed 'tar' archives.

    'src3r2.tar.Z':         the *source code* for the GAP  kernel.  You  need
                            this unless you get one of the executables below.
                            This file is about 750 KBytes long.

    'lib3r2.tar.Z':         the *library of functions*.  You need this.  This
                            file is about 1000 KBytes long.

    'doc3r2.tar.Z':         the  *documentation*.  Serves  as  LaTeX   source
                            for the printed manual and  online documentation.
                            Contains further installation  information.  This
                            file is about 850 KBytes long.

    'doc3r2.dvi.Z':         the preformatted  documentation.  You  need  this
                            if you do not have a  *big*  TeX.  This  file  is
                            about 1100 KByte long.

    'grp3r2.tar.Z':         various *group libraries*.  Contains for  example
                            all primitive permutation  groups  of  degree  at
                            most 50.  This file is about 50 KByte long.

    'two3r2.tar.Z':         the library of *2-group* of  size  at  most  256.
                            This file is about 650 KByte long.

    'thr3r2.tar.Z':         the library of *3-groups* of  size at  most  729.
                            This file is about 20 KByte long.

    'tbl3r2.tar.Z':         a library of *character tables* including all  of
                            the ATLAS.  This file is about 2050 KByte long.

    'tom3r2.tar.Z':         a library of *table of marks* of various  groups.
                            This file is about 450 KByte long.

    'anupq.tar.Z':          the *ANU PQ* share library package.  This file is
                            about 350 KByte long.

    'nq.tar.Z':             the *NQ*  share  library  package.  This  file is
                            about 100 KByte long.

    'weyl.tar.Z':           the *Weyl* share library package.  This  file  is
                            about 50 KByte long.

    'src3r2.zoo', 'lib3r2.zoo', 'doc3r2.zoo', 'grp3r2.zoo'
    'tbl3r2.zoo', 'two3r2.zoo', 'thr3r2.zoo', 'tom3r2.zoo',
    'anupq.zoo',  'nq.zoo',     'weyl.zoo':
                            'zoo'  archives  containing  *exactly*  the  same
                            files as the 'compress'-ed 'tar' archives  above.
                            The advantage of 'compress'-ed 'tar'  archives is
                            that  'uncompress' and 'tar' are widely available
                            on UNIX systems.  The advantage of 'zoo' archives
                            is  that they are smaller (about  30 percent) and
                            that 'zoo' is more common on PC-s and Atari ST-s.
                            (These files may not be available on all servers)

    We  supply  executables  for  machines  that don't usually  come with a C
    compiler or machines where  the  standard  C  compiler does  not  produce
    optimal results.  If you have one of those machines it will be easier for
    you  to  get  this executable  instead  of compiling  GAP  yourself.  The
    following  executables  are  available  (again  these  files  may not  be
    available on all servers)

    'gapexe.386':           executable for IBM PC compatibles with  an  Intel
                            80386 or  80486 running MS-DOS 5.0 compiled  with
                            the GNU  C  2.2.2  compiler.   See below  for the
                            copyright.  This file is about 500 KByte long.

    'gapexe.next':          executable  for the NeXT (680?0) running NeXTstep
                            3.0  compiled  with  GNU C 2.3.3 compiler.   This
                            file is about 400 KByte long.

    'gapexe.st':            executable  for  Atari  ST  (680?0)  running  TOS
                            compiled with the GNU C compiler.  This  file  is
                            about 450 KByte long.

    'gapexe.su3':           executable for SUN 3 (680?0) running SunOS 4.0 or
                            higher compiled  with the  GNU C  compiler.  This
                            file is about 500 KByte long.

    'gapexe.su4':           executable for SUN 4 (Sparc) running SunOS 4.1 or
                            higher  compiled with the GNU  C  2.3.2 compiler.
                            This file is about 600 KByte long.

    The following support files are also available (and again these files may
    not be available on all servers)

    'compress.tar':         'compress' version 4.1.  You  need  this  program
                            to uncompress  the  compressed  tar  files.  Note
                            however, that almost all UNIX systems  these days
                            already come with an executable 'compress'.  This
                            file is about 90 KByte long.

    'patch.tar.Z':          Larry  Wall's  'patch'  program  version  2.0.2.0
                            (patchlevel 12u4).  This program  can  be used to
                            automatically  apply upgrades.   Note  that older
                            versions  of 'patch' are *not* able to understand
                            the  unified 'diff'  format  used  in the upgrade
                            files.  This file is about 70 KByte long.

    'uud.c':                'uud' version 3.4.  'uud' is much better than the
                            'uudecode' that comes  with  most  UNIX  systems.
                            This file is about 12 KByte long.

    'zoo21.tar.Z':          Rahul Dhesi's  'zoo' archiver  version  2.1.  You
                            need this  to  unpack  the  *zoo-archives*.  Note
                            that the widespread version 2.01 will *not* work.
                            This file is about 250 KByte long.

    'zooexe.386':           Executable of 'zoo' for IBM PC compatibles.  This
                            file is about 55 KByte long.

    'zooexe.st':            Executable of 'zoo' for the Atari ST.  This  file
                            is about 80 KByte long.


How to install GAP
==================

    The file  'install.tex' in 'doc3r2.tar.Z' contains extensive installation
    instructions.  If however, you are one of those  who never  read manuals,
    here is a quick installation guide.

    First for UNIX.

    Make a directory for GAP,  e.g., '~/gap/'  or  '/usr/local/lib/gap/'.

    Unpack the source  archive  'src3r2.tar.Z' into the subdirectory  'src/';
    unpack the library archive  'lib3r2.tar.Z' into the subdirectory  'lib/';
    unpack the documentation    'doc3r2.tar.Z' into the subdirectory  'doc/'.

    If you  have obtained the optional groups and character tables  libraries
    'grp3r2.tar.Z',   'tbl3r2.tar.Z',  'two3r2.tar.Z',   'thr3r2.tar.Z',  and
    'tom3r2.tar.Z',  unpack  them  into the  subdirectories  'grp/',  'tbl/',
    'two/', 'thr/', or 'tom/'.

    Change into 'src/' and execute 'make' to  see a list of possible targets;
    select a target, if in doubt use 'bsd' or 'usg', and make the kernel.

    In an appropriate directory, e.g., '~/bin/' or  '/usr/local/bin/', create
    a shell script that executes the GAP kernel.  This should look like

       exec <gap-directory>/src/gap  -m 4m  -l <gap-directory>/lib/  $*

    The option '-m' specifies the amount  of initial memory; the  option '-l'
    specifies where to find the library, if you get it wrong GAP complains

       gap: hmm, I cannot find 'lib/init.g', maybe use option '-l <libname>'?

    Change into 'doc/' and make the printed manual with the commands

       latex manual;  latex manual;  lp -dvi manual.dvi

    or something similar, according to your local custom for using LaTeX.

    Try something in GAP,  e.g., the following exercises GAP quite a bit

       gap> m11 := Group( (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6) );;
       gap> Number( ConjugacyClasses( m11 ) );

    The result should be 10.

    Next for IBM PC compatibles with an Intel 80386 or 80486 running  MS-DOS.

    Make a directory for GAP, e.g, 'c:\gap\'.

    Put the executable 'gapexe.386' into this directory calling it 'gap.exe'.
    Unpack the library archive   'lib3r2.zoo'  into the subdirectory  'lib\';
    unpack the documentation     'doc3r2.zoo'  into the subdirectory  'doc\'.

    If  you have  obtained the optional  groups and character table libraries
    'grp3r2.zoo', 'tbl3r2.zoo', 'two3r2.zoo', 'thr3r2.zoo', and 'tom3r2.zoo',
    unpack  them into the subdirectories 'grp\', 'tbl\', 'two\',  'thr\', and
    'tom\'.

    In a directory in  your  path,  e.g.,  'c:\bin\',  create  a  batch  file
    'gap.bat' that executes the GAP kernel.  This should look like

       <gap-directory>\gap  -m 4m  -l <gap-directory>\lib\  %1 %2 %3 %4

    The option '-m' specifies the amount  of initial memory; the  option '-l'
    specifies where to find the library, if you get it wrong GAP complains

       gap: hmm, I cannot find 'lib/init.g', maybe use option '-l <libname>'?

    Add the following line to your 'autoexec.bat' file

       SET GO32TMP=<swap-file-directory>

    where <swap-file-directory> should be the directory where you want GAP to
    put the  swap  file,  e.g.,  'c:\tmp'.   The swap  file  will  be  called
    'page????.386' and  is normally removed when GAP exits.  If 'GO32TMP'  is
    not set, 'GCCTMP', 'TMP', 'TEMP' are checked (in this order).  If neither
    is  set, GAP  will not swap to  disk.  *Note  that you must reboot before
    this change in 'autoexec.bat' takes effect*.

    Change into 'doc\' and make the printed manual with the commands

       latex manual;  latex manual;  print manual.dvi

    or something similar, according to your local custom for using LaTeX.

    Try something in GAP,  e.g., the following exercises GAP quite a bit

       gap> m11 := Group( (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6) );;
       gap> Number( ConjugacyClasses( m11 ) );

    The result should be 10.

    Note that GAP for the 386  will  use up  to 128 MByte of extended  memory
    (using XMS, VDISK memory  allocation  strategies)  or up to 128  MByte of
    expanded memory (using VCPI programs, such as QEMM  and 386MAX) and up to
    128 MByte of disk space for swapping.   Further note that GAP for the 386
    will *not* run under Windows (because it does not support DPMI).

    If you hit <ctr>-'C' the DOS  extender ('go32') catches it and aborts GAP
    immediately.  The  keys  <ctr>-'Z' and  <alt>-'C'  can be used instead to
    interupt GAP.

    The arrow keys <left>, <right>, <up>, <down>, <home>, <end>, and <delete>
    can be used for command line editing with their intuitive meaning.

    Pathnames may be  given inside GAP using either shlash ('/') or backslash
    ('\') as a separator (though '\' must be escaped in strings of course).

    The system dependent  part of GAP for the 386 ('sysdos.c') was written by
    Steve Linton (111  Ross  St.,  Cambridge,  CB1 3BS, UK,  +44 223  411661,
    'sl25@cus.cam.ac.uk').  He assignes the copyright to the Lehrstuhl D fuer
    Mathematik.  Many thanks to Steve Linton for his work.

    GAP for the 386 was compiled  with DJ Delorie's port of the Free Software
    Foundation's GNU C compiler version 2.1.  The compiler can be obtained by
    anonymous  'ftp'  from  'grape.ecs.clarkson.edu'  where   it  is  in  the
    directory 'pub/msdos/djgpp'.  Many thanks to the Free Software Foundation
    and DJ Delorie for this amazing piece of work.

    The GNU C compiler is

        Copyright (C) 1989 Free Software Foundation, Inc.
                           675 Mass Ave, Cambridge, MA 02139, USA

    under the  terms  of the GNU General Public License (GPL).  Note that the
    GNU  GPL  states  that the  mere act  of compiling  does not  affect  the
    copyright status of GAP.

    The modifications to the compiler to make it operating under  MS-DOS, the
    functions from  the standard library 'libpc.a', the  modifications of the
    functions from the standard library  'libc.a' to make  them operate under
    MS-DOS, and the DOS extender  'go32' (which is prepended to 'gapexe.386')
    are

        Copyright (C) 1991 DJ Delorie,
                           24 Kirsten Ave, Rochester NH 03867-2954, USA

    also under  the terms of the GNU GPL.  The terms of  the GPL require that
    we make the source code for 'libpc.a' available.  They can be obtained by
    writing to Steve Linton (however, it may be  easier for you to 'ftp' them
    from  'grape.ecs.clarkson.edu'  yourself).   They  also require that  GAP
    falls under the GPL too, i.e., is distributed free   coof users.  We also hope that this number will be  large enough
    to convince various  agencies that GAP is a project worthy of (financial)
    support.   If you publish some result that was partly obtained using GAP,
    we would  appreciate  it  if you  would cite GAP,  just as you would cite
    another paper that you  used.  Again  we  would  appreciate if  you could
    inform us about such a paper.

    We  distribute the  *full  source*  for  everything, the  C code for  the
    kernel, the GAP code for the library, and the LaTeX  code for the manual,
    which has  at present about 800 pages.  So it should be no problem to get
    GAP,  even if you have a rather uncommon system.  Of course, ports to non
    UNIX systems may require some work.   We already  have ports  for  IBM PC
    compatibles with an Intel 80386 or 80486 and for  the Atari ST.   We also
    hope to provide a port  of  GAP 3.2 to the Apple  Macintosh  in the  near
    future (there is already  a port of GAP 3.1).  Note that about 4 MByte of
    main memory and a harddisk are required to run GAP.

    GAP 3.2 can be obtained by anonymous *ftp* from the following servers.

    'samson.math.rwth-aachen.de':
            Lehrstuhl D fur Mathematik, RWTH Aachen, Germany (137.226.152.6).

    'dimacs.rutgers.edu':
            DIMACS, Rutgers, New Brunswick, New Jersey (128.6.75.16).

    'math.ucla.edu':
            Math. Dept., Univ. of California at Los Angeles (128.97.4.254).

    'wuarchive.wustl.edu':
    	    Mathematics Archives, Univ. of Tennessee (128.252.135.4,
            directory '/edu/math/source.code/group.theory/gap').

    'pell.anu.edu.au':
            Math. Research Section, Australian National Univ. (150.203.15.5).

    'ftp' to the server  *closest* to  you, login as user 'ftp' and give your
    full  e-mail  address  as password.  GAP  is in the  directory 'pub/gap'.
    Remember when you transmit  the files  to set the  file  transfer type to
    *binary image*, otherwise you will only receive unusable  garbage.  Those
    servers will always have the latest version of GAP available.

    GAP can also be obtained via *electronic mail*.  To get  one of the files
    mentioned  below  send a message to 'listserv@samson.math.rwth-aachen.de'
    containing a line  'get GAP <file-name>',  e.g., 'get  GAP src3r2.tar.Z'.
    'listserv' will reply by sending you the file as e-mail message.

    Because most files are  large  binary files they will  be  uuencoded  and
    split into  several parts,  each   at  most 64  kBytes  large.   You  can
    concatenate  the parts  by hand,  removing the mail  header, and then use
    'uudecode' to decode them.  We suggest however that you also get 'uud.c',
    which skips  the  mail headers automatically  and  is also able to fix up
    transmission errors caused by 'EBCDIC' machines.  You can also get single
    parts of a file by sending 'get GAP <file-name> <part-nr>'.

    For users in the United Kingdom with only Janet access, neither 'ftp' nor
    the mail server will work (please do *not*  try to  use the mail server).
    Please contact Derek Holt (e-mail address 'dfh@maths.warwick.ac.uk').  He
    has kindly offered us to distribute GAP in the United Kingdom.

    The 'ftp'  directory and  the 'listserv' archive   contain  the following
    files.  Please check first  which files you  need, to  avoid transferring
    those that you don't need.

    'README':               the file you are currently reading.

    GAP version 3 release 2 itself  comes in several files.   You do not need
    all of those files.  All files are 'compress'-ed 'tar' archives.

    'src3r2.tar.Z':         the *source code* for the GAP  kernel.  You  need
                            this unless you get one of the executables below.
                            This file is about 750 KBytes long.

    'lib3r2.tar.Z':         the *library of functions*.  You need this.  This
                            file is about 1000 KBytes long.

    'doc3r2.tar.Z':         the  *documentation*.  Serves  as  LaTeX   source
                            for the printed manual and  online documentation.
                            Contains further installation  information.  This
                            file is about 850 KBytes long.

    'doc3r2.dvi.Z':         the preformatted  documentation.  You  need  this
                            if you do not have a  *big*  TeX.  This  file  is
                            about 1100 KByte long.

    'grp3r2.tar.Z':         various *group libraries*.  Contains for  example
                            all primitive permutation  groups  of  degree  at
                            most 50.  This file is about 50 KByte long.

    'two3r2.tar.Z':         the library of *2-group* of  size  at  most  256.
                            This file is about 650 KByte long.

    'thr3r2.tar.Z':         the library of *3-groups* of  size at  most  729.
                            This file is about 20 KByte long.

    'tbl3r2.tar.Z':         a library of *character tables* including all  of
                            the ATLAS.  This file is about 2050 KByte long.

    'tom3r2.tar.Z':         a library of *table of marks* of various  groups.
                            This file is about 450 KByte long.

    'anupq.tar.Z':          the *ANU PQ* share library package.  This file is
                            about 350 KByte long.

    'nq.tar.Z':             the *NQ*  share  library  package.  This  file is
                            about 100 KByte long.

    'weyl.tar.Z':           the *Weyl* share library package.  This  file  is
                            about 50 KByte long.

    'src3r2.zoo', 'lib3r2.zoo', 'doc3r2.zoo', 'grp3r2.zoo'
    'tbl3r2.zoo', 'two3r2.zoo', 'thr3r2.zoo', 'tom3r2.zoo',
    'anupq.zoo',  'nq.zoo',     'weyl.zoo':
                            'zoo'  archives  containing  *exactly*  the  same
                            files as the 'compress'-ed 'tar' archives  above.
                            The advantage of 'compress'-ed 'tar'  archives is
                            that  'uncompress' and 'tar' are widely available
                            on UNIX systems.  The advantage of 'zoo' archives
                            is  that they are smaller (about  30 percent) and
                            that 'zoo' is more common on PC-s and Atari ST-s.
                            (These files may not be available on all servers)

    We  supply  executables  for  machines  that don't usually  come with a C
    compiler or machines where  the  standard  C  compiler does  not  produce
    optimal results.  If you have one of those machines it will be easier for
    you  to  get  this executable  instead  of compiling  GAP  yourself.  The
    following  executables  are  available  (again  these  files  may not  be
    available on all servers)

    'gapexe.386':           executable for IBM PC compatibles with  an  Intel
                            80386 or  80486 running MS-DOS 5.0 compiled  with
                            the GNU  C  2.2.2  compiler.   See below  for the
                            copyright.  This file is about 500 KByte long.

    'gapexe.next':          executable  for the NeXT (680?0) running NeXTstep
                            3.0  compiled  with  GNU C 2.3.3 compiler.   This
                            file is about 400 KByte long.

    'gapexe.st':            executable  for  Atari  ST  (680?0)  running  TOS
                            compiled with the GNU C compiler.  This  file  is
                            about 450 KByte long.

    'gapexe.su3':           executable for SUN 3 (680?0) running SunOS 4.0 or
                            higher compiled  with the  GNU C  compiler.  This
                            file is about 500 KByte long.

    'gapexe.su4':           executable for SUN 4 (Sparc) running SunOS 4.1 or
                            higher  compiled with the GNU  C  2.3.2 compiler.
                            This file is about 600 KByte long.

    The following support files are also available (and again these files may
    not be available on all servers)

    'compress.tar':         'compress' version 4.1.  You  need  this  program
                            to uncompress  the  compressed  tar  files.  Note
                            however, that almost all UNIX systems  these days
                            already come with an executable 'compress'.  This
                            file is about 90 KByte long.

    'patch.tar.Z':          Larry  Wall's  'patch'  program  version  2.0.2.0
                            (patchlevel 12u4).  This program  can  be used to
                            automatically  apply upgrades.   Note  that older
                            versions  of 'patch' are *not* able to understand
                            the  unified 'diff'  format  used  in the upgrade
                            files.  This file is about 70 KByte long.

    'uud.c':                'uud' version 3.4.  'uud' is much better than the
                            'uudecode' that comes  with  most  UNIX  systems.
                            This file is about 12 KByte long.

    'zoo21.tar.Z':          Rahul Dhesi's  'zoo' archiver  version  2.1.  You
                            need this  to  unpack  the  *zoo-archives*.  Note
                            that the widespread version 2.01 will *not* work.
                            This file is about 250 KByte long.

    'zooexe.386':           Executable of 'zoo' for IBM PC compatibles.  This
                            file is about 55 KByte long.

    'zooexe.st':            Executable of 'zoo' for the Atari ST.  This  file
                            is about 80 KByte long.


How to install GAP
==================

    The file  'install.tex' in 'doc3r2.tar.Z' contains extensive installation
    instructions.  If however, you are one of those  who never  read manuals,
    here is a quick installation guide.

    First for UNIX.

    Make a directory for GAP,  e.g., '~/gap/'  or  '/usr/local/lib/gap/'.

    Unpack the source  archive  'src3r2.tar.Z' into the subdirectory  'src/';
    unpack the library archive  'lib3r2.tar.Z' into the subdirectory  'lib/';
    unpack the documentation    'doc3r2.tar.Z' into the subdirectory  'doc/'.

    If you  have obtained the optional groups and character tables  libraries
    'grp3r2.tar.Z',   'tbl3r2.tar.Z',  'two3r2.tar.Z',   'thr3r2.tar.Z',  and
    'tom3r2.tar.Z',  unpack  them  into the  subdirectories  'grp/',  'tbl/',
    'two/', 'thr/', or 'tom/'.

    Change into 'src/' and execute 'make' to  see a list of possible targets;
    select a target, if in doubt use 'bsd' or 'usg', and make the kernel.

    In an appropriate directory, e.g., '~/bin/' or  '/usr/local/bin/', create
    a shell script that executes the GAP kernel.  This should look like

       exec <gap-directory>/src/gap  -m 4m  -l <gap-directory>/lib/  $*

    The option '-m' specifies the amount  of initial memory; the  option '-l'
    specifies where to find the library, if you get it wrong GAP complains

       gap: hmm, I cannot find 'lib/init.g', maybe use option '-l <libname>'?

    Change into 'doc/' and make the printed manual with the commands

       latex manual;  latex manual;  lp -dvi manual.dvi

    or something similar, according to your local custom for using LaTeX.

    Try something in GAP,  e.g., the following exercises GAP quite a bit

       gap> m11 := Group( (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6) );;
       gap> Number( ConjugacyClasses( m11 ) );

    The result should be 10.

    Next for IBM PC compatibles with an Intel 80386 or 80486 running  MS-DOS.

    Make a directory for GAP, e.g, 'c:\gap\'.

    Put the executable 'gapexe.386' into this directory calling it 'gap.exe'.
    Unpack the library archive   'lib3r2.zoo'  into the subdirectory  'lib\';
    unpack the documentation     'doc3r2.zoo'  into the subdirectory  'doc\'.

    If  you have  obtained the optional  groups and character table libraries
    'grp3r2.zoo', 'tbl3r2.zoo', 'two3r2.zoo', 'thr3r2.zoo', and 'tom3r2.zoo',
    unpack  them into the subdirectories 'grp\', 'tbl\', 'two\',  'thr\', and
    'tom\'.

    In a directory in  your  path,  e.g.,  'c:\bin\',  create  a  batch  file
    'gap.bat' that executes the GAP kernel.  This should look like

       <gap-directory>\gap  -m 4m  -l <gap-directory>\lib\  %1 %2 %3 %4

    The option '-m' specifies the amount  of initial memory; the  option '-l'
    specifies where to find the library, if you get it wrong GAP complains

       gap: hmm, I cannot find 'lib/init.g', maybe use option '-l <libname>'?

    Add the following line to your 'autoexec.bat' file

       SET GO32TMP=<swap-file-directory>

    where <swap-file-directory> should be the directory where you want GAP to
    put the  swap  file,  e.g.,  'c:\tmp'.   The swap  file  will  be  called
    'page????.386' and  is normally removed when GAP exits.  If 'GO32TMP'  is
    not set, 'GCCTMP', 'TMP', 'TEMP' are checked (in this order).  If neither
    is  set, GAP  will not swap to  disk.  *Note  that you must reboot before
    this change in 'autoexec.bat' takes effect*.

    Change into 'doc\' and make the printed manual with the commands

       latex manual;  latex manual;  print manual.dvi

    or something similar, according to your local custom for using LaTeX.

    Try something in GAP,  e.g., the following exercises GAP quite a bit

       gap> m11 := Group( (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6) );;
       gap> Number( ConjugacyClasses( m11 ) );

    The result should be 10.

    Note that GAP for the 386  will  use up  to 128 MByte of extended  memory
    (using XMS, VDISK memory  allocation  strategies)  or up to 128  MByte of
    expanded memory (using VCPI programs, such as QEMM  and 386MAX) and up to
    128 MByte of disk space for swapping.   Further note that GAP for the 386
    will *not* run under Windows (because it does not support DPMI).

    If you hit <ctr>-'C' the DOS  extender ('go32') catches it and aborts GAP
    immediately.  The  keys  <ctr>-'Z' and  <alt>-'C'  can be used instead to
    interupt GAP.

    The arrow keys <left>, <right>, <up>, <down>, <home>, <end>, and <delete>
    can be used for command line editing with their intuitive meaning.

    Pathnames may be  given inside GAP using either shlash ('/') or backslash
    ('\') as a separator (though '\' must be escaped in strings of course).

    The system dependent  part of GAP for the 386 ('sysdos.c') was written by
    Steve Linton (111  Ross  St.,  Cambridge,  CB1 3BS, UK,  +44 223  411661,
    'sl25@cus.cam.ac.uk').  He assignes the copyright to the Lehrstuhl D fuer
    Mathematik.  Many thanks to Steve Linton for his work.

    GAP for the 386 was compiled  with DJ Delorie's port of the Free Software
    Foundation's GNU C compiler version 2.1.  The compiler can be obtained by
    anonymous  'ftp'  from  'grape.ecs.clarkson.edu'  where   it  is  in  the
    directory 'pub/msdos/djgpp'.  Many thanks to the Free Software Foundation
    and DJ Delorie for this amazing piece of work.

    The GNU C compiler is

        Copyright (C) 1989 Free Software Foundation, Inc.
                           675 Mass Ave, Cambridge, MA 02139, USA

    under the  terms  of the GNU General Public License (GPL).  Note that the
    GNU  GPL  states  that the  mere act  of compiling  does not  affect  the
    copyright status of GAP.

    The modifications to the compiler to make it operating under  MS-DOS, the
    functions from  the standard library 'libpc.a', the  modifications of the
    functions from the standard library  'libc.a' to make  them operate under
    MS-DOS, and the DOS extender  'go32' (which is prepended to 'gapexe.386')
    are

        Copyright (C) 1991 DJ Delorie,
                           24 Kirsten Ave, Rochester NH 03867-2954, USA

    also under  the terms of the GNU GPL.  The terms of  the GPL require that
    we make the source code for 'libpc.a' available.  They can be obtained by
    writing to Steve Linton (however, it may be  easier for you to 'ftp' them
    from  'grape.ecs.clarkson.edu'  yourself).   They  also require that  GAP
    falls under the GPL too, i.e., is distributed free   coof users.  We also hope that this number will be  large enough
    to convince various  agencies that GAP is a project worthy of (financial)
    support.   If you publish some result that was partly obtained using GAP,
    we would  appreciate  it  if you  would cite GAP,  just as you would cite
    another paper that you  used.  Again  we  would  appreciate if  you could
    inform us about such a paper.

    We  distribute the  *full  source*  for  everything, the  C code for  the
    kernel, the GAP code for the library, and the LaTeX  code for the manual,
    which has  at present about 800 pages.  So it should be no problem to get
    GAP,  even if you have a rather uncommon system.  Of course, ports to non
    UNIX systems may require some work.   We already  have ports  for  IBM PC
    compatibles with an Intel 80386 or 80486 and for  the Atari ST.   We also
    hope to provide a port  of  GAP 3.2 to the Apple  Macintosh  in the  near
    future (there is already  a port of GAP 3.1).  Note that about 4 MByte of
    main memory and a harddisk are required to run GAP.

    GAP 3.2 can be obtained by anonymous *ftp* from the following servers.

    'samson.math.rwth-aachen.de':
            Lehrstuhl D fur Mathematik, RWTH Aachen, Germany (137.226.152.6).

    'dimacs.rutgers.edu':
            DIMACS, Rutgers, New Brunswick, New Jersey (128.6.75.16).

    'math.ucla.edu':
            Math. Dept., Univ. of California at Los Angeles (128.97.4.254).

    'wuarchive.wustl.edu':
    	    Mathematics Archives, Univ. of Tennessee (128.252.135.4,
            directory '/edu/math/source.code/group.theory/gap').

    'pell.anu.edu.au':
            Math. Research Section, Australian National Univ. (150.203.15.5).

    'ftp' to the server  *closest* to  you, login as user 'ftp' and give your
    full  e-mail  address  as password.  GAP  is in the  directory 'pub/gap'.
    Remember when you transmit  the files  to set the  file  transfer type to
    *binary image*, otherwise you will only receive unusable  garbage.  Those
    servers will always have the latest version of GAP available.

    GAP can also be obtained via *electronic mail*.  To get  one of the files
    mentioned  below  send a message to 'listserv@samson.math.rwth-aachen.de'
    containing a line  'get GAP <file-name>',  e.g., 'get  GAP src3r2.tar.Z'.
    'listserv' will reply by sending you the file as e-mail message.

    Because most files are  large  binary files they will  be  uuencoded  and
    split into  several parts,  each   at  most 64  kBytes  large.   You  can
    concatenate  the parts  by hand,  removing the mail  header, and then use
    'uudecode' to decode them.  We suggest however that you also get 'uud.c',
    which skips  the  mail headers automatically  and  is also able to fix up
    transmission errors caused by 'EBCDIC' machines.  You can also get single
    parts of a file by sending 'get GAP <file-name> <part-nr>'.

    For users in the United Kingdom with only Janet access, neither 'ftp' nor
    the mail server will work (please do *not*  try to  use the mail server).
    Please contact Derek Holt (e-mail address 'dfh@maths.warwick.ac.uk').  He
    has kindly offered us to distribute GAP in the United Kingdom.

    The 'ftp'  directory and  the 'listserv' archive   contain  the following
    files.  Please check first  which files you  need, to  avoid transferring
    those that you don't need.

    'README':               the file you are currently reading.

    GAP version 3 release 2 itself  comes in several files.   You do not need
    all of those files.  All files are 'compress'-ed 'tar' archives.

    'src3r2.tar.Z':         the *source code* for the GAP  kernel.  You  need
                            this unless you get one of the executables below.
                            This file is about 750 KBytes long.

    'lib3r2.tar.Z':         the *library of functions*.  You need this.  This
                            file is about 1000 KBytes long.

    'doc3r2.tar.Z':         the  *documentation*.  Serves  as  LaTeX   source
                            for the printed manual and  online documentation.
                            Contains further installation  information.  This
                            file is about 850 KBytes long.

    'doc3r2.dvi.Z':         the preformatted  documentation.  You  need  this
                            if you do not have a  *big*  TeX.  This  file  is
                            about 1100 KByte long.

    'grp3r2.tar.Z':         various *group libraries*.  Contains for  example
                            all primitive permutation  groups  of  degree  at
                            most 50.  This file is about 50 KByte long.

    'two3r2.tar.Z':         the library of *2-group* of  size  at  most  256.
                            This file is about 650 KByte long.

    'thr3r2.tar.Z':         the library of *3-groups* of  size at  most  729.
                            This file is about 20 KByte long.

    'tbl3r2.tar.Z':         a library of *character tables* including all  of
                            the ATLAS.  This file is about 2050 KByte long.

    'tom3r2.tar.Z':         a library of *table of marks* of various  groups.
                            This file is about 450 KByte long.

    'anupq.tar.Z':          the *ANU PQ* share library package.  This file is
                            about 350 KByte long.

    'nq.tar.Z':             the *NQ*  share  library  package.  This  file is
                            about 100 KByte long.

    'weyl.tar.Z':           the *Weyl* share library package.  This  file  is
                            about 50 KByte long.

    'src3r2.zoo', 'lib3r2.zoo', 'doc3r2.zoo', 'grp3r2.zoo'
    'tbl3r2.zoo', 'two3r2.zoo', 'thr3r2.zoo', 'tom3r2.zoo',
    'anupq.zoo',  'nq.zoo',     'weyl.zoo':
                            'zoo'  archives  containing  *exactly*  the  same
                            files as the 'compress'-ed 'tar' archives  above.
                            The advantage of 'compress'-ed 'tar'  archives is
                            that  'uncompress' and 'tar' are widely available
                            on UNIX systems.  The advantage of 'zoo' archives
                            is  that they are smaller (about  30 percent) and
                            that 'zoo' is more common on PC-s and Atari ST-s.
                            (These files may not be available on all servers)

    We  supply  executables  for  machines  that don't usually  come with a C
    compiler or machines where  the  standard  C  compiler does  not  produce
    optimal results.  If you have one of those machines it will be easier for
    you  to  get  this executable  instead  of compiling  GAP  yourself.  The
    following  executables  are  available  (again  these  files  may not  be
    available on all servers)

    'gapexe.386':           executable for IBM PC compatibles with  an  Intel
                            80386 or  80486 running MS-DOS 5.0 compiled  with
                            the GNU  C  2.2.2  compiler.   See below  for the
                            copyright.  This file is about 500 KByte long.

    'gapexe.next':          executable  for the NeXT (680?0) running NeXTstep
                            3.0  compiled  with  GNU C 2.3.3 compiler.   This
                            file is about 400 KByte long.

    'gapexe.st':            executable  for  Atari  ST  (680?0)  running  TOS
                            compiled with the GNU C compiler.  This  file  is
                            about 450 KByte long.

    'gapexe.su3':           executable for SUN 3 (680?0) running SunOS 4.0 or
                            higher compiled  with the  GNU C  compiler.  This
                            file is about 500 KByte long.

    'gapexe.su4':           executable for SUN 4 (Sparc) running SunOS 4.1 or
                            higher  compiled with the GNU  C  2.3.2 compiler.
                            This file is about 600 KByte long.

    The following support files are also available (and again these files may
    not be available on all servers)

    'compress.tar':         'compress' version 4.1.  You  need  this  program
                            to uncompress  the  compressed  tar  files.  Note
                            however, that almost all UNIX systems  these days
                            already come with an executable 'compress'.  This
                            file is about 90 KByte long.

    'patch.tar.Z':          Larry  Wall's  'patch'  program  version  2.0.2.0
                            (patchlevel 12u4).  This program  can  be used to
                            automatically  apply upgrades.   Note  that older
                            versions  of 'patch' are *not* able to understand
                            the  unified 'diff'  format  used  in the upgrade
                            files.  This file is about 70 KByte long.

    'uud.c':                'uud' version 3.4.  'uud' is much better than the
                            'uudecode' that comes  with  most  UNIX  systems.
                            This file is about 12 KByte long.

    'zoo21.tar.Z':          Rahul Dhesi's  'zoo' archiver  version  2.1.  You
                            need this  to  unpack  the  *zoo-archives*.  Note
                            that the widespread version 2.01 will *not* work.
                            This file is about 250 KByte long.

    'zooexe.386':           Executable of 'zoo' for IBM PC compatibles.  This
                            file is about 55 KByte long.

    'zooexe.st':            Executable of 'zoo' for the Atari ST.  This  file
                            is about 80 KByte long.


How to install GAP
==================

    The file  'install.tex' in 'doc3r2.tar.Z' contains extensive installation
    instructions.  If however, you are one of those  who never  read manuals,
    here is a quick installation guide.

    First for UNIX.

    Make a directory for GAP,  e.g., '~/gap/'  or  '/usr/local/lib/gap/'.

    Unpack the source  archive  'src3r2.tar.Z' into the subdirectory  'src/';
    unpack the library archive  'lib3r2.tar.Z' into the subdirectory  'lib/';
    unpack the documentation    'doc3r2.tar.Z' into the subdirectory  'doc/'.

    If you  have obtained the optional groups and character tables  libraries
    'grp3r2.tar.Z',   'tbl3r2.tar.Z',  'two3r2.tar.Z',   'thr3r2.tar.Z',  and
    'tom3r2.tar.Z',  unpack  them  into the  subdirectories  'grp/',  'tbl/',
    'two/', 'thr/', or 'tom/'.

    Change into 'src/' and execute 'make' to  see a list of possible targets;
    select a target, if in doubt use 'bsd' or 'usg', and make the kernel.

    In an appropriate directory, e.g., '~/bin/' or  '/usr/local/bin/', create
    a shell script that executes the GAP kernel.  This should look like

       exec <gap-directory>/src/gap  -m 4m  -l <gap-directory>/lib/  $*

    The option '-m' specifies the amount  of initial memory; the  option '-l'
    specifies where to find the library, if you get it wrong GAP complains

       gap: hmm, I cannot find 'lib/init.g', maybe use option '-l <libname>'?

    Change into 'doc/' and make the printed manual with the commands

       latex manual;  latex manual;  lp -dvi manual.dvi

    or something similar, according to your local custom for using LaTeX.

    Try something in GAP,  e.g., the following exercises GAP quite a bit

       gap> m11 := Group( (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6) );;
       gap> Number( ConjugacyClasses( m11 ) );

    The result should be 10.

    Next for IBM PC compatibles with an Intel 80386 or 80486 running  MS-DOS.

    Make a directory for GAP, e.g, 'c:\gap\'.

    Put the executable 'gapexe.386' into this directory calling it 'gap.exe'.
    Unpack the library archive   'lib3r2.zoo'  into the subdirectory  'lib\';
    unpack the documentation     'doc3r2.zoo'  into the subdirectory  'doc\'.

    If  you have  obtained the optional  groups and character table libraries
    'grp3r2.zoo', 'tbl3r2.zoo', 'two3r2.zoo', 'thr3r2.zoo', and 'tom3r2.zoo',
    unpack  them into the subdirectories 'grp\', 'tbl\', 'two\',  'thr\', and
    'tom\'.

    In a directory in  your  path,  e.g.,  'c:\bin\',  create  a  batch  file
    'gap.bat' that executes the GAP kernel.  This should look like

       <gap-directory>\gap  -m 4m  -l <gap-directory>\lib\  %1 %2 %3 %4

    The option '-m' specifies the amount  of initial memory; the  option '-l'
    specifies where to find the library, if you get it wrong GAP complains

       gap: hmm, I cannot find 'lib/init.g', maybe use option '-l <libname>'?

    Add the following line to your 'autoexec.bat' file

       SET GO32TMP=<swap-file-directory>

    where <swap-file-directory> should be the directory where you want GAP to
    put the  swap  file,  e.g.,  'c:\tmp'.   The swap  file  will  be  called
    'page????.386' and  is normally removed when GAP exits.  If 'GO32TMP'  is
    not set, 'GCCTMP', 'TMP', 'TEMP' are checked (in this order).  If neither
    is  set, GAP  will not swap to  disk.  *Note  that you must reboot before
    this change in 'autoexec.bat' takes effect*.

    Change into 'doc\' and make the printed manual with the commands

       latex manual;  latex manual;  print manual.dvi

    or something similar, according to your local custom for using LaTeX.

    Try something in GAP,  e.g., the following exercises GAP quite a bit

       gap> m11 := Group( (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6) );;
       gap> Number( ConjugacyClasses( m11 ) );

    The result should be 10.

    Note that GAP for the 386  will  use up  to 128 MByte of extended  memory
    (using XMS, VDISK memory  allocation  strategies)  or up to 128  MByte of
    expanded memory (using VCPI programs, such as QEMM  and 386MAX) and up to
    128 MByte of disk space for swapping.   Further note that GAP for the 386
    will *not* run under Windows (because it does not support DPMI).

    If you hit <ctr>-'C' the DOS  extender ('go32') catches it and aborts GAP
    immediately.  The  keys  <ctr>-'Z' and  <alt>-'C'  can be used instead to
    interupt GAP.

    The arrow keys <left>, <right>, <up>, <down>, <home>, <end>, and <delete>
    can be used for command line editing with their intuitive meaning.

    Pathnames may be  given inside GAP using either shlash ('/') or backslash
    ('\') as a separator (though '\' must be escaped in strings of course).

    The system dependent  part of GAP for the 386 ('sysdos.c') was written by
    Steve Linton (111  Ross  St.,  Cambridge,  CB1 3BS, UK,  +44 223  411661,
    'sl25@cus.cam.ac.uk').  He assignes the copyright to the Lehrstuhl D fuer
    Mathematik.  Many thanks to Steve Linton for his work.

    GAP for the 386 was compiled  with DJ Delorie's port of the Free Software
    Foundation's GNU C compiler version 2.1.  The compiler can be obtained by
    anonymous  'ftp'  from  'grape.ecs.clarkson.edu'  where   it  is  in  the
    directory 'pub/msdos/djgpp'.  Many thanks to the Free Software Foundation
    and DJ Delorie for this amazing piece of work.

    The GNU C compiler is

        Copyright (C) 1989 Free Software Foundation, Inc.
                           675 Mass Ave, Cambridge, MA 02139, USA

    under the  terms  of the GNU General Public License (GPL).  Note that the
    GNU  GPL  states  that the  mere act  of compiling  does not  affect  the
    copyright status of GAP.

    The modifications to the compiler to make it operating under  MS-DOS, the
    functions from  the standard library 'libpc.a', the  modifications of the
    functions from the standard library  'libc.a' to make  them operate under
    MS-DOS, and the DOS extender  'go32' (which is prepended to 'gapexe.386')
    are

        Copyright (C) 1991 DJ Delorie,
                           24 Kirsten Ave, Rochester NH 03867-2954, USA

    also under  the terms of the GNU GPL.  The terms of  the GPL require that
    we make the source code for 'libpc.a' available.  They can be obtained by
    writing to Steve Linton (however, it may be  easier for you to 'ftp' them
    from  'grape.ecs.clarkson.edu'  yourself).   They  also require that  GAP
    falls under the GPL too, i.e., is distributed free   coof users.  We also hope that this number will be  large enough
    to convince various  agencies that GAP is a project worthy of (financial)
    support.   If you publish some result that was partly obtained using GAP,
    we would  appreciate  it  if you  would cite GAP,  just as you would cite
    another paper that you  used.  Again  we  would  appreciate if  you could
    inform us about such a paper.

    We  distribute the  *full  source*  for  everything, the  C code for  the
    kernel, the GAP code for the library, and the LaTeX  code for the manual,
    which has  at present about 800 pages.  So it should be no problem to get
    GAP,  even if you have a rather uncommon system.  Of course, ports to non
    UNIX systems may require some work.   We already  have ports  for  IBM PC
    compatibles with an Intel 80386 or 80486 and for  the Atari ST.   We also
    hope to provide a port  of  GAP 3.2 to the Apple  Macintosh  in the  near
    future (there is already  a port of GAP 3.1).  Note that about 4 MByte of
    main memory and a harddisk are required to run GAP.

    GAP 3.2 can be obtained by anonymous *ftp* from the following servers.

    'samson.math.rwth-aachen.de':
            Lehrstuhl D fur Mathematik, RWTH Aachen, Germany (137.226.152.6).

    'dimacs.rutgers.edu':
            DIMACS, Rutgers, New Brunswick, New Jersey (128.6.75.16).

    'math.ucla.edu':
            Math. Dept., Univ. of California at Los Angeles (128.97.4.254).

    'wuarchive.wustl.edu':
    	    Mathematics Archives, Univ. of Tennessee (128.252.135.4,
            directory '/edu/math/source.code/group.theory/gap').

    'pell.anu.edu.au':
            Math. Research Section, Australian National Univ. (150.203.15.5).

    'ftp' to the server  *closest* to  you, login as user 'ftp' and give your
    full  e-mail  address  as password.  GAP  is in the  directory 'pub/gap'.
    Remember when you transmit  the files  to set the  file  transfer type to
    *binary image*, otherwise you will only receive unusable  garbage.  Those
    servers will always have the latest version of GAP available.

    GAP can also be obtained via *electronic mail*.  To get  one of the files
    mentioned  below  send a message to 'listserv@samson.math.rwth-aachen.de'
    containing a line  'get GAP <file-name>',  e.g., 'get  GAP src3r2.tar.Z'.
    'listserv' will reply by sending you the file as e-mail message.

    Because most files are  large  binary files they will  be  uuencoded  and
    split into  several parts,  each   at  most 64  kBytes  large.   You  can
    concatenate  the parts  by hand,  removing the mail  header, and then use
    'uudecode' to decode them.  We suggest however that you also get 'uud.c',
    which skips  the  mail headers automatically  and  is also able to fix up
    transmission errors caused by 'EBCDIC' machines.  You can also get single
    parts of a file by sending 'get GAP <file-name> <part-nr>'.

    For users in the United Kingdom with only Janet access, neither 'ftp' nor
    the mail server will work (please do *not*  try to  use the mail server).
    Please contact Derek Holt (e-mail address 'dfh@maths.warwick.ac.uk').  He
    has kindly offered us to distribute GAP in the United Kingdom.

    The 'ftp'  directory and  the 'listserv' archive   contain  the following
    files.  Please check first  which files you  need, to  avoid transferring
    those that you don't need.

    'README':               the file you are currently reading.

    GAP version 3 release 2 itself  comes in several files.   You do not need
    all of those files.  All files are 'compress'-ed 'tar' archives.

    'src3r2.tar.Z':         the *source code* for the GAP  kernel.  You  need
                            this unless you get one of the executables below.
                            This file is about 750 KBytes long.

    'lib3r2.tar.Z':         the *library of functions*.  You need this.  This
                            file is about 1000 KBytes long.

    'doc3r2.tar.Z':         the  *documentation*.  Serves  as  LaTeX   source
                            for the printed manual and  online documentation.
                            Contains further installation  information.  This
                            file is about 850 KBytes long.

    'doc3r2.dvi.Z':         the preformatted  documentation.  You  need  this
                            if you do not have a  *big*  TeX.  This  file  is
                            about 1100 KByte long.

    'grp3r2.tar.Z':         various *group libraries*.  Contains for  example
                            all primitive permutation  groups  of  degree  at
                            most 50.  This file is about 50 KByte long.

    'two3r2.tar.Z':         the library of *2-group* of  size  at  most  256.
                            This file is about 650 KByte long.

    'thr3r2.tar.Z':         the library of *3-groups* of  size at  most  729.
                            This file is about 20 KByte long.

    'tbl3r2.tar.Z':         a library of *character tables* including all  of
                            the ATLAS.  This file is about 2050 KByte long.

    'tom3r2.tar.Z':         a library of *table of marks* of various  groups.
                            This file is about 450 KByte long.

    'anupq.tar.Z':          the *ANU PQ* share library package.  This file is
                            about 350 KByte long.

    'nq.tar.Z':             the *NQ*  share  library  package.  This  file is
                            about 100 KByte long.

    'weyl.tar.Z':           the *Weyl* share library package.  This  file  is
                            about 50 KByte long.

    'src3r2.zoo', 'lib3r2.zoo', 'doc3r2.zoo', 'grp3r2.zoo'
    'tbl3r2.zoo', 'two3r2.zoo', 'thr3r2.zoo', 'tom3r2.zoo',
    'anupq.zoo',  'nq.zoo',     'weyl.zoo':
                            'zoo'  archives  containing  *exactly*  the  same
                            files as the 'compress'-ed 'tar' archives  above.
                            The advantage of 'compress'-ed 'tar'  archives is
                            that  'uncompress' and 'tar' are widely available
                            on UNIX systems.  The advantage of 'zoo' archives
                            is  that they are smaller (about  30 percent) and
                            that 'zoo' is more common on PC-s and Atari ST-s.
                            (These files may not be available on all servers)

    We  supply  executables  for  machines  that don't usually  come with a C
    compiler or machines where  the  standard  C  compiler does  not  produce
    optimal results.  If you have one of those machines it will be easier for
    you  to  get  this executable  instead  of compiling  GAP  yourself.  The
    following  executables  are  available  (again  these  files  may not  be
    available on all servers)

    'gapexe.386':           executable for IBM PC compatibles with  an  Intel
                            80386 or  80486 running MS-DOS 5.0 compiled  with
                            the GNU  C  2.2.2  compiler.   See below  for the
                            copyright.  This file is about 500 KByte long.

    'gapexe.next':          executable  for the NeXT (680?0) running NeXTstep
                            3.0  compiled  with  GNU C 2.3.3 compiler.   This
                            file is about 400 KByte long.

    'gapexe.st':            executable  for  Atari  ST  (680?0)  running  TOS
                            compiled with the GNU C compiler.  This  file  is
                            about 450 KByte long.

    'gapexe.su3':           executable for SUN 3 (680?0) running SunOS 4.0 or
                            higher compiled  with the  GNU C  compiler.  This
                            file is about 500 KByte long.

    'gapexe.su4':           executable for SUN 4 (Sparc) running SunOS 4.1 or
                            higher  compiled with the GNU  C  2.3.2 compiler.
                            This file is about 600 KByte long.

    The following support files are also available (and again these files may
    not be available on all servers)

    'compress.tar':         'compress' version 4.1.  You  need  this  program
                            to uncompress  the  compressed  tar  files.  Note
                            however, that almost all UNIX systems  these days
                            already come with an executable 'compress'.  This
                            file is about 90 KByte long.

    'patch.tar.Z':          Larry  Wall's  'patch'  program  version  2.0.2.0
                            (patchlevel 12u4).  This program  can  be used to
                            automatically  apply upgrades.   Note  that older
                            versions  of 'patch' are *not* able to understand
                            the  unified 'diff'  format  used  in the upgrade
                            files.  This file is about 70 KByte long.

    'uud.c':                'uud' version 3.4.  'uud' is much better than the
                            'uudecode' that comes  with  most  UNIX  systems.
                            This file is about 12 KByte long.

    'zoo21.tar.Z':          Rahul Dhesi's  'zoo' archiver  version  2.1.  You
                            need this  to  unpack  the  *zoo-archives*.  Note
                            that the widespread version 2.01 will *not* work.
                            This file is about 250 KByte long.

    'zooexe.386':           Executable of 'zoo' for IBM PC compatibles.  This
                            file is about 55 KByte long.

    'zooexe.st':            Executable of 'zoo' for the Atari ST.  This  file
                            is about 80 KByte long.


How to install GAP
==================

    The file  'install.tex' in 'doc3r2.tar.Z' contains extensive installation
    instructions.  If however, you are one of those  who never  read manuals,
    here is a quick installation guide.

    First for UNIX.

    Make a directory for GAP,  e.g., '~/gap/'  or  '/usr/local/lib/gap/'.

    Unpack the source  archive  'src3r2.tar.Z' into the subdirectory  'src/';
    unpack the library archive  'lib3r2.tar.Z' into the subdirectory  'lib/';
    unpack the documentation    'doc3r2.tar.Z' into the subdirectory  'doc/'.

    If you  have obtained the optional groups and character tables  libraries
    'grp3r2.tar.Z',   'tbl3r2.tar.Z',  'two3r2.tar.Z',   'thr3r2.tar.Z',  and
    'tom3r2.tar.Z',  unpack  them  into the  subdirectories  'grp/',  'tbl/',
    'two/', 'thr/', or 'tom/'.

    Change into 'src/' and execute 'make' to  see a list of possible targets;
    select a target, if in doubt use 'bsd' or 'usg', and make the kernel.

    In an appropriate directory, e.g., '~/bin/' or  '/usr/local/bin/', create
    a shell script that executes the GAP kernel.  This should look like

       exec <gap-directory>/src/gap  -m 4m  -l <gap-directory>/lib/  $*

    The option '-m' specifies the amount  of initial memory; the  option '-l'
    specifies where to find the library, if you get it wrong GAP complains

       gap: hmm, I cannot find 'lib/init.g', maybe use option '-l <libname>'?

    Change into 'doc/' and make the printed manual with the commands

       latex manual;  latex manual;  lp -dvi manual.dvi

    or something similar, according to your local custom for using LaTeX.

    Try something in GAP,  e.g., the following exercises GAP quite a bit

       gap> m11 := Group( (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6) );;
       gap> Number( ConjugacyClasses( m11 ) );

    The result should be 10.

    Next for IBM PC compatibles with an Intel 80386 or 80486 running  MS-DOS.

    Make a directory for GAP, e.g, 'c:\gap\'.

    Put the executable 'gapexe.386' into this directory calling it 'gap.exe'.
    Unpack the library archive   'lib3r2.zoo'  into the subdirectory  'lib\';
    unpack the documentation     'doc3r2.zoo'  into the subdirectory  'doc\'.

    If  you have  obtained the optional  groups and character table libraries
    'grp3r2.zoo', 'tbl3r2.zoo', 'two3r2.zoo', 'thr3r2.zoo', and 'tom3r2.zoo',
    unpack  them into the subdirectories 'grp\', 'tbl\', 'two\',  'thr\', and
    'tom\'.

    In a directory in  your  path,  e.g.,  'c:\bin\',  create  a  batch  file
    'gap.bat' that executes the GAP kernel.  This should look like

       <gap-directory>\gap  -m 4m  -l <gap-directory>\lib\  %1 %2 %3 %4

    The option '-m' specifies the amount  of initial memory; the  option '-l'
    specifies where to find the library, if you get it wrong GAP complains

       gap: hmm, I cannot find 'lib/init.g', maybe use option '-l <libname>'?

    Add the following line to your 'autoexec.bat' file

       SET GO32TMP=<swap-file-directory>

    where <swap-file-directory> should be the directory where you want GAP to
    put the  swap  file,  e.g.,  'c:\tmp'.   The swap  file  will  be  called
    'page????.386' and  is normally removed when GAP exits.  If 'GO32TMP'  is
    not set, 'GCCTMP', 'TMP', 'TEMP' are checked (in this order).  If neither
    is  set, GAP  will not swap to  disk.  *Note  that you must reboot before
    this change in 'autoexec.bat' takes effect*.

    Change into 'doc\' and make the printed manual with the commands

       latex manual;  latex manual;  print manual.dvi

    or something similar, according to your local custom for using LaTeX.

    Try something in GAP,  e.g., the following exercises GAP quite a bit

       gap> m11 := Group( (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6) );;
       gap> Number( ConjugacyClasses( m11 ) );

    The result should be 10.

    Note that GAP for the 386  will  use up  to 128 MByte of extended  memory
    (using XMS, VDISK memory  allocation  strategies)  or up to 128  MByte of
    expanded memory (using VCPI programs, such as QEMM  and 386MAX) and up to
    128 MByte of disk space for swapping.   Further note that GAP for the 386
    will *not* run under Windows (because it does not support DPMI).

    If you hit <ctr>-'C' the DOS  extender ('go32') catches it and aborts GAP
    immediately.  The  keys  <ctr>-'Z' and  <alt>-'C'  can be used instead to
    interupt GAP.

    The arrow keys <left>, <right>, <up>, <down>, <home>, <end>, and <delete>
    can be used for command line editing with their intuitive meaning.

    Pathnames may be  given inside GAP using either shlash ('/') or backslash
    ('\') as a separator (though '\' must be escaped in strings of course).

    The system dependent  part of GAP for the 386 ('sysdos.c') was written by
    Steve Linton (111  Ross  St.,  Cambridge,  CB1 3BS, UK,  +44 223  411661,
    'sl25@cus.cam.ac.uk').  He assignes the copyright to the Lehrstuhl D fuer
    Mathematik.  Many thanks to Steve Linton for his work.

    GAP for the 386 was compiled  with DJ Delorie's port of the Free Software
    Foundation's GNU C compiler version 2.1.  The compiler can be obtained by
    anonymous  'ftp'  from  'grape.ecs.clarkson.edu'  where   it  is  in  the
    directory 'pub/msdos/djgpp'.  Many thanks to the Free Software Foundation
    and DJ Delorie for this amazing piece of work.

    The GNU C compiler is

        Copyright (C) 1989 Free Software Foundation, Inc.
                           675 Mass Ave, Cambridge, MA 02139, USA

    under the  terms  of the GNU General Public License (GPL).  Note that the
    GNU  GPL  states  that the  mere act  of compiling  does not  affect  the
    copyright status of GAP.

    The modifications to the compiler to make it operating under  MS-DOS, the
    functions from  the standard library 'libpc.a', the  modifications of the
    functions from the standard library  'libc.a' to make  them operate under
    MS-DOS, and the DOS extender  'go32' (which is prepended to 'gapexe.386')
    are

        Copyright (C) 1991 DJ Delorie,
                           24 Kirsten Ave, Rochester NH 03867-2954, USA

    also under  the terms of the GNU GPL.  The terms of  the GPL require that
    we make the source code for 'libpc.a' available.  They can be obtained by
    writing to Steve Linton (however, it may be  easier for you to 'ftp' them
    from  'grape.ecs.clarkson.edu'  yourself).   They  also require that  GAP
    falls under the GPL too, i.e., is distributed free   coof users.  We also hope that this number will be  large enough
    to convince various  agencies that GAP is a project worthy of (financial)
    support.   If you publish some result that was partly obtained using GAP,
    we would  appreciate  it  if you  would cite GAP,  just as you would cite
    another paper that you  used.  Again  we  would  appreciate if  you could
    inform us about such a paper.

    We  distribute the  *full  source*  for  everything, the  C code for  the
    kernel, the GAP code for the library, and the LaTeX  code for the manual,
    which has  at present about 800 pages.  So it should be no problem to get
    GAP,  even if you have a rather uncommon system.  Of course, ports to non
    UNIX systems may require some work.   We already  have ports  for  IBM PC
    compatibles with an Intel 80386 or 80486 and for  the Atari ST.   We also
    hope to provide a port  of  GAP 3.2 to the Apple  Macintosh  in the  near
    future (there is already  a port of GAP 3.1).  Note that about 4 MByte of
    main memory and a harddisk are required to run GAP.

    GAP 3.2 can be obtained by anonymous *ftp* from the following servers.

    'samson.math.rwth-aachen.de':
            Lehrstuhl D fur Mathematik, RWTH Aachen, Germany (137.226.152.6).

    'dimacs.rutgers.edu':
            DIMACS, Rutgers, New Brunswick, New Jersey (128.6.75.16).

    'math.ucla.edu':
            Math. Dept., Univ. of California at Los Angeles (128.97.4.254).

    'wuarchive.wustl.edu':
    	    Mathematics Archives, Univ. of Tennessee (128.252.135.4,
            directory '/edu/math/source.code/group.theory/gap').

    'pell.anu.edu.au':
            Math. Research Section, Australian National Univ. (150.203.15.5).

    'ftp' to the server  *closest* to  you, login as user 'ftp' and give your
    full  e-mail  address  as password.  GAP  is in the  directory 'pub/gap'.
    Remember when you transmit  the files  to set the  file  transfer type to
    *binary image*, otherwise you will only receive unusable  garbage.  Those
    servers will always have the latest version of GAP available.

    GAP can also be obtained via *electronic mail*.  To get  one of the files
    mentioned  below  send a message to 'listserv@samson.math.rwth-aachen.de'
    containing a line  'get GAP <file-name>',  e.g., 'get  GAP src3r2.tar.Z'.
    'listserv' will reply by sending you the file as e-mail message.

    Because most files are  large  binary files they will  be  uuencoded  and
    split into  several parts,  each   at  most 64  kBytes  large.   You  can
    concatenate  the parts  by hand,  removing the mail  header, and then use
    'uudecode' to decode them.  We suggest however that you also get 'uud.c',
    which skips  the  mail headers automatically  and  is also able to fix up
    transmission errors caused by 'EBCDIC' machines.  You can also get single
    parts of a file by sending 'get GAP <file-name> <part-nr>'.

    For users in the United Kingdom with only Janet access, neither 'ftp' nor
    the mail server will work (please do *not*  try to  use the mail server).
    Please contact Derek Holt (e-mail address 'dfh@maths.warwick.ac.uk').  He
    has kindly offered us to distribute GAP in the United Kingdom.

    The 'ftp'  directory and  the 'listserv' archive   contain  the following
    files.  Please check first  which files you  need, to  avoid transferring
    those that you don't need.

    'README':               the file you are currently reading.

    GAP version 3 release 2 itself  comes in several files.   You do not need
    all of those files.  All files are 'compress'-ed 'tar' archives.

    'src3r2.tar.Z':         the *source code* for the GAP  kernel.  You  need
                            this unless you get one of the executables below.
                            This file is about 750 KBytes long.

    'lib3r2.tar.Z':         the *library of functions*.  You need this.  This
                            file is about 1000 KBytes long.

    'doc3r2.tar.Z':         the  *documentation*.  Serves  as  LaTeX   source
                            for the printed manual and  online documentation.
                            Contains further installation  information.  This
                            file is about 850 KBytes long.

    'doc3r2.dvi.Z':         the preformatted  documentation.  You  need  this
                            if you do not have a  *big*  TeX.  This  file  is
                            about 1100 KByte long.

    'grp3r2.tar.Z':         various *group libraries*.  Contains for  example
                            all primitive permutation  groups  of  degree  at
                            most 50.  This file is about 50 KByte long.

    'two3r2.tar.Z':         the library of *2-group* of  size  at  most  256.
                            This file is about 650 KByte long.

    'thr3r2.tar.Z':         the library of *3-groups* of  size at  most  729.
                            This file is about 20 KByte long.

    'tbl3r2.tar.Z':         a library of *character tables* including all  of
                            the ATLAS.  This file is about 2050 KByte long.

    'tom3r2.tar.Z':         a library of *table of marks* of various  groups.
                            This file is about 450 KByte long.

    'anupq.tar.Z':          the *ANU PQ* share library package.  This file is
                            about 350 KByte long.

    'nq.tar.Z':             the *NQ*  share  library  package.  This  file is
                            about 100 KByte long.

    'weyl.tar.Z':           the *Weyl* share library package.  This  file  is
                            about 50 KByte long.

    'src3r2.zoo', 'lib3r2.zoo', 'doc3r2.zoo', 'grp3r2.zoo'
    'tbl3r2.zoo', 'two3r2.zoo', 'thr3r2.zoo', 'tom3r2.zoo',
    'anupq.zoo',  'nq.zoo',     'weyl.zoo':
                            'zoo'  archives  containing  *exactly*  the  same
                            files as the 'compress'-ed 'tar' archives  above.
                            The advantage of 'compress'-ed 'tar'  archives is
                            that  'uncompress' and 'tar' are widely available
                            on UNIX systems.  The advantage of 'zoo' archives
                            is  that they are smaller (about  30 percent) and
                            that 'zoo' is more common on PC-s and Atari ST-s.
                            (These files may not be available on all servers)

    We  supply  executables  for  machines  that don't usually  come with a C
    compiler or machines where  the  standard  C  compiler does  not  produce
    optimal results.  If you have one of those machines it will be easier for
    you  to  get  this executable  instead  of compiling  GAP  yourself.  The
    following  executables  are  available  (again  these  files  may not  be
    available on all servers)

    'gapexe.386':           executable for IBM PC compatibles with  an  Intel
                            80386 or  80486 running MS-DOS 5.0 compiled  with
                            the GNU  C  2.2.2  compiler.   See below  for the
                            copyright.  This file is about 500 KByte long.

    'gapexe.next':          executable  for the NeXT (680?0) running NeXTstep
                            3.0  compiled  with  GNU C 2.3.3 compiler.   This
                            file is about 400 KByte long.

    'gapexe.st':            executable  for  Atari  ST  (680?0)  running  TOS
                            compiled with the GNU C compiler.  This  file  is
                            about 450 KByte long.

    'gapexe.su3':           executable for SUN 3 (680?0) running SunOS 4.0 or
                            higher compiled  with the  GNU C  compiler.  This
                            file is about 500 KByte long.

    'gapexe.su4':           executable for SUN 4 (Sparc) running SunOS 4.1 or
                            higher  compiled with the GNU  C  2.3.2 compiler.
                            This file is about 600 KByte long.

    The following support files are also available (and again these files may
    not be available on all servers)

    'compress.tar':         'compress' version 4.1.  You  need  this  program
                            to uncompress  the  compressed  tar  files.  Note
                            however, that almost all UNIX systems  these days
                            already come with an executable 'compress'.  This
                            file is about 90 KByte long.

    'patch.tar.Z':          Larry  Wall's  'patch'  program  version  2.0.2.0
                            (patchlevel 12u4).  This program  can  be used to
                            automatically  apply upgrades.   Note  that older
                            versions  of 'patch' are *not* able to understand
                            the  unified 'diff'  format  used  in the upgrade
                            files.  This file is about 70 KByte long.

    'uud.c':                'uud' version 3.4.  'uud' is much better than the
                            'uudecode' that comes  with  most  UNIX  systems.
                            This file is about 12 KByte long.

    'zoo21.tar.Z':          Rahul Dhesi's  'zoo' archiver  version  2.1.  You
                            need this  to  unpack  the  *zoo-archives*.  Note
                            that the widespread version 2.01 will *not* work.
                            This file is about 250 KByte long.

    'zooexe.386':           Executable of 'zoo' for IBM PC compatibles.  This
                            file is about 55 KByte long.

    'zooexe.st':            Executable of 'zoo' for the Atari ST.  This  file
                            is about 80 KByte long.


How to install GAP
==================

    The file  'install.tex' in 'doc3r2.tar.Z' contains extensive installation
    instructions.  If however, you are one of those  who never  read manuals,
    here is a quick installation guide.

    First for UNIX.

    Make a directory for GAP,  e.g., '~/gap/'  or  '/usr/local/lib/gap/'.

    Unpack the source  archive  'src3r2.tar.Z' into the subdirectory  'src/';
    unpack the library archive  'lib3r2.tar.Z' into the subdirectory  'lib/';
    unpack the documentation    'doc3r2.tar.Z' into the subdirectory  'doc/'.

    If you  have obtained the optional groups and character tables  libraries
    'grp3r2.tar.Z',   'tbl3r2.tar.Z',  'two3r2.tar.Z',   'thr3r2.tar.Z',  and
    'tom3r2.tar.Z',  unpack  them  into the  subdirectories  'grp/',  'tbl/',
    'two/', 'thr/', or 'tom/'.

    Change into 'src/' and execute 'make' to  see a list of possible targets;
    select a target, if in doubt use 'bsd' or 'usg', and make the kernel.

    In an appropriate directory, e.g., '~/bin/' or  '/usr/local/bin/', create
    a shell script that executes the GAP kernel.  This should look like

       exec <gap-directory>/src/gap  -m 4m  -l <gap-directory>/lib/  $*

    The option '-m' specifies the amount  of initial memory; the  option '-l'
    specifies where to find the library, if you get it wrong GAP complains

       gap: hmm, I cannot find 'lib/init.g', maybe use option '-l <libname>'?

    Change into 'doc/' and make the printed manual with the commands

       latex manual;  latex manual;  lp -dvi manual.dvi

    or something similar, according to your local custom for using LaTeX.

    Try something in GAP,  e.g., the following exercises GAP quite a bit

       gap> m11 := Group( (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6) );;
       gap> Number( ConjugacyClasses( m11 ) );

    The result should be 10.

    Next for IBM PC compatibles with an Intel 80386 or 80486 running  MS-DOS.

    Make a directory for GAP, e.g, 'c:\gap\'.

    Put the executable 'gapexe.386' into this directory calling it 'gap.exe'.
    Unpack the library archive   'lib3r2.zoo'  into the subdirectory  'lib\';
    unpack the documentation     'doc3r2.zoo'  into the subdirectory  'doc\'.

    If  you have  obtained the optional  groups and character table libraries
    'grp3r2.zoo', 'tbl3r2.zoo', 'two3r2.zoo', 'thr3r2.zoo', and 'tom3r2.zoo',
    unpack  them into the subdirectories 'grp\', 'tbl\', 'two\',  'thr\', and
    'tom\'.

    In a directory in  your  path,  e.g.,  'c:\bin\',  create  a  batch  file
    'gap.bat' that executes the GAP kernel.  This should look like

       <gap-directory>\gap  -m 4m  -l <gap-directory>\lib\  %1 %2 %3 %4

    The option '-m' specifies the amount  of initial memory; the  option '-l'
    specifies where to find the library, if you get it wrong GAP complains

       gap: hmm, I cannot find 'lib/init.g', maybe use option '-l <libname>'?

    Add the following line to your 'autoexec.bat' file

       SET GO32TMP=<swap-file-directory>

    where <swap-file-directory> should be the directory where you want GAP to
    put the  swap  file,  e.g.,  'c:\tmp'.   The swap  file  will  be  called
    'page????.386' and  is normally removed when GAP exits.  If 'GO32TMP'  is
    not set, 'GCCTMP', 'TMP', 'TEMP' are checked (in this order).  If neither
    is  set, GAP  will not swap to  disk.  *Note  that you must reboot before
    this change in 'autoexec.bat' takes effect*.

    Change into 'doc\' and make the printed manual with the commands

       latex manual;  latex manual;  print manual.dvi

    or something similar, according to your local custom for using LaTeX.

    Try something in GAP,  e.g., the following exercises GAP quite a bit

       gap> m11 := Group( (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6) );;
       gap> Number( ConjugacyClasses( m11 ) );

    The result should be 10.

    Note that GAP for the 386  will  use up  to 128 MByte of extended  memory
    (using XMS, VDISK memory  allocation  strategies)  or up to 128  MByte of
    expanded memory (using VCPI programs, such as QEMM  and 386MAX) and up to
    128 MByte of disk space for swapping.   Further note that GAP for the 386
    will *not* run under Windows (because it does not support DPMI).

    If you hit <ctr>-'C' the DOS  extender ('go32') catches it and aborts GAP
    immediately.  The  keys  <ctr>-'Z' and  <alt>-'C'  can be used instead to
    interupt GAP.

    The arrow keys <left>, <right>, <up>, <down>, <home>, <end>, and <delete>
    can be used for command line editing with their intuitive meaning.

    Pathnames may be  given inside GAP using either shlash ('/') or backslash
    ('\') as a separator (though '\' must be escaped in strings of course).

    The system dependent  part of GAP for the 386 ('sysdos.c') was written by
    Steve Linton (111  Ross  St.,  Cambridge,  CB1 3BS, UK,  +44 223  411661,
    'sl25@cus.cam.ac.uk').  He assignes the copyright to the Lehrstuhl D fuer
    Mathematik.  Many thanks to Steve Linton for his work.

    GAP for the 386 was compiled  with DJ Delorie's port of the Free Software
    Foundation's GNU C compiler version 2.1.  The compiler can be obtained by
    anonymous  'ftp'  from  'grape.ecs.clarkson.edu'  where   it  is  in  the
    directory 'pub/msdos/djgpp'.  Many thanks to the Free Software Foundation
    and DJ Delorie for this amazing piece of work.

    The GNU C compiler is

        Copyright (C) 1989 Free Software Foundation, Inc.
                           675 Mass Ave, Cambridge, MA 02139, USA

    under the  terms  of the GNU General Public License (GPL).  Note that the
    GNU  GPL  states  that the  mere act  of compiling  does not  affect  the
    copyright status of GAP.

    The modifications to the compiler to make it operating under  MS-DOS, the
    functions from  the standard library 'libpc.a', the  modifications of the
    functions from the standard library  'libc.a' to make  them operate under
    MS-DOS, and the DOS extender  'go32' (which is prepended to 'gapexe.386')
    are

        Copyright (C) 1991 DJ Delorie,
                           24 Kirsten Ave, Rochester NH 03867-2954, USA

    also under  the terms of the GNU GPL.  The terms of  the GPL require that
    we make the source code for 'libpc.a' available.  They can be obtained by
    writing to Steve Linton (however, it may be  easier for you to 'ftp' them
    from  'grape.ecs.clarkson.edu'  yourself).   They  also require that  GAP
    falls under the GPL too, i.e., is distributed free   coof users.  We also hope that this number will be  large enough
    to convince various  agencies that GAP is a project worthy of (financial)
    support.   If you publish some result that was partly obtained using GAP,
    we would  appreciate  it  if you  would cite GAP,  just as you would cite
    another paper that you  used.  Again  we  would  appreciate if  you could
    inform us about such a paper.

    We  distribute the  *full  source*  for  everything, the  C code for  the
    kernel, the GAP code for the library, and the LaTeX  code for the manual,
    which has  at present about 800 pages.  So it should be no problem to get
    GAP,  even if you have a rather uncommon system.  Of course, ports to non
    UNIX systems may require some work.   We already  have ports  for  IBM PC
    compatibles with an Intel 80386 or 80486 and for  the Atari ST.   We also
    hope to provide a port  of  GAP 3.2 to the Apple  Macintosh  in the  near
    future (there is already  a port of GAP 3.1).  Note that about 4 MByte of
    main memory and a harddisk are required to run GAP.

    GAP 3.2 can be obtained by anonymous *ftp* from the following servers.

    'samson.math.rwth-aachen.de':
            Lehrstuhl D fur Mathematik, RWTH Aachen, Germany (137.226.152.6).

    'dimacs.rutgers.edu':
            DIMACS, Rutgers, New Brunswick, New Jersey (128.6.75.16).

    'math.ucla.edu':
            Math. Dept., Univ. of California at Los Angeles (128.97.4.254).

    'wuarchive.wustl.edu':
    	    Mathematics Archives, Univ. of Tennessee (128.252.135.4,
            directory '/edu/math/source.code/group.theory/gap').

    'pell.anu.edu.au':
            Math. Research Section, Australian National Univ. (150.203.15.5).

    'ftp' to the server  *closest* to  you, login as user 'ftp' and give your
    full  e-mail  address  as password.  GAP  is in the  directory 'pub/gap'.
    Remember when you transmit  the files  to set the  file  transfer type to
    *binary image*, otherwise you will only receive unusable  garbage.  Those
    servers will always have the latest version of GAP available.

    GAP can also be obtained via *electronic mail*.  To get  one of the files
    mentioned  below  send a message to 'listserv@samson.math.rwth-aachen.de'
    containing a line  'get GAP <file-name>',  e.g., 'get  GAP src3r2.tar.Z'.
    'listserv' will reply by sending you the file as e-mail message.

    Because most files are  large  binary files they will  be  uuencoded  and
    split into  several parts,  each   at  most 64  kBytes  large.   You  can
    concatenate  the parts  by hand,  removing the mail  header, and then use
    'uudecode' to decode them.  We suggest however that you also get 'uud.c',
    which skips  the  mail headers automatically  and  is also able to fix up
    transmission errors caused by 'EBCDIC' machines.  You can also get single
    parts of a file by sending 'get GAP <file-name> <part-nr>'.

    For users in the United Kingdom with only Janet access, neither 'ftp' nor
    the mail server will work (please do *not*  try to  use the mail server).
    Please contact Derek Holt (e-mail address 'dfh@maths.warwick.ac.uk').  He
    has kindly offered us to distribute GAP in the United Kingdom.

    The 'ftp'  directory and  the 'listserv' archive   contain  the following
    files.  Please check first  which files you  need, to  avoid transferring
    those that you don't need.

    'README':               the file you are currently reading.

    GAP version 3 release 2 itself  comes in several files.   You do not need
    all of those files.  All files are 'compress'-ed 'tar' archives.

    'src3r2.tar.Z':         the *source code* for the GAP  kernel.  You  need
                            this unless you get one of the executables below.
                            This file is about 750 KBytes long.

    'lib3r2.tar.Z':         the *library of functions*.  You need this.  This
                            file is about 1000 KBytes long.

    'doc3r2.tar.Z':         the  *documentation*.  Serves  as  LaTeX   source
                            for the printed manual and  online documentation.
                            Contains further installation  information.  This
                            file is about 850 KBytes long.

    'doc3r2.dvi.Z':         the preformatted  documentation.  You  need  this
                            if you do not have a  *big*  TeX.  This  file  is
                            about 1100 KByte long.

    'grp3r2.tar.Z':         various *group libraries*.  Contains for  example
                            all primitive permutation  groups  of  degree  at
                            most 50.  This file is about 50 KByte long.

    'two3r2.tar.Z':         the library of *2-group* of  size  at  most  256.
                            This file is about 650 KByte long.

    'thr3r2.tar.Z':         the library of *3-groups* of  size at  most  729.
                            This file is about 20 KByte long.

    'tbl3r2.tar.Z':         a library of *character tables* including all  of
                            the ATLAS.  This file is about 2050 KByte long.

    'tom3r2.tar.Z':         a library of *table of marks* of various  groups.
                            This file is about 450 KByte long.

    'anupq.tar.Z':          the *ANU PQ* share library package.  This file is
                            about 350 KByte long.

    'nq.tar.Z':             the *NQ*  share  library  package.  This  file is
                            about 100 KByte long.

    'weyl.tar.Z':           the *Weyl* share library package.  This  file  is
                            about 50 KByte long.

    'src3r2.zoo', 'lib3r2.zoo', 'doc3r2.zoo', 'grp3r2.zoo'
    'tbl3r2.zoo', 'two3r2.zoo', 'thr3r2.zoo', 'tom3r2.zoo',
    'anupq.zoo',  'nq.zoo',     'weyl.zoo':
                            'zoo'  archives  containing  *exactly*  the  same
                            files as the 'compress'-ed 'tar' archives  above.
                            The advantage of 'compress'-ed 'tar'  archives is
                            that  'uncompress' and 'tar' are widely available
                            on UNIX systems.  The advantage of 'zoo' archives
                            is  that they are smaller (about  30 percent) and
                            that 'zoo' is more common on PC-s and Atari ST-s.
                            (These files may not be available on all servers)

    We  supply  executables  for  machines  that don't usually  come with a C
    compiler or machines where  the  standard  C  compiler does  not  produce
    optimal results.  If you have one of those machines it will be easier for
    you  to  get  this executable  instead  of compiling  GAP  yourself.  The
    following  executables  are  available  (again  these  files  may not  be
    available on all servers)

    'gapexe.386':           executable for IBM PC compatibles with  an  Intel
                            80386 or  80486 running MS-DOS 5.0 compiled  with
                            the GNU  C  2.2.2  compiler.   See below  for the
                            copyright.  This file is about 500 KByte long.

    'gapexe.next':          executable  for the NeXT (680?0) running NeXTstep
                            3.0  compiled  with  GNU C 2.3.3 compiler.   This
                            file is about 400 KByte long.

    'gapexe.st':            executable  for  Atari  ST  (680?0)  running  TOS
                            compiled with the GNU C compiler.  This  file  is
                            about 450 KByte long.

    'gapexe.su3':           executable for SUN 3 (680?0) running SunOS 4.0 or
                            higher compiled  with the  GNU C  compiler.  This
                            file is about 500 KByte long.

    'gapexe.su4':           executable for SUN 4 (Sparc) running SunOS 4.1 or
                            higher  compiled with the GNU  C  2.3.2 compiler.
                            This file is about 600 KByte long.

    The following support files are also available (and again these files may
    not be available on all servers)

    'compress.tar':         'compress' version 4.1.  You  need  this  program
                            to uncompress  the  compressed  tar  files.  Note
                            however, that almost all UNIX systems  these days
                            already come with an executable 'compress'.  This
                            file is about 90 KByte long.

    'patch.tar.Z':          Larry  Wall's  'patch'  program  version  2.0.2.0
                            (patchlevel 12u4).  This program  can  be used to
                            automatically  apply upgrades.   Note  that older
                            versions  of 'patch' are *not* able to understand
                            the  unified 'diff'  format  used  in the upgrade
                            files.  This file is about 70 KByte long.

    'uud.c':                'uud' version 3.4.  'uud' is much better than the
                            'uudecode' that comes  with  most  UNIX  systems.
                            This file is about 12 KByte long.

    'zoo21.tar.Z':          Rahul Dhesi's  'zoo' archiver  version  2.1.  You
                            need this  to  unpack  the  *zoo-archives*.  Note
                            that the widespread version 2.01 will *not* work.
                            This file is about 250 KByte long.

    'zooexe.386':           Executable of 'zoo' for IBM PC compatibles.  This
                            file is about 55 KByte long.

    'zooexe.st':            Executable of 'zoo' for the Atari ST.  This  file
                            is about 80 KByte long.


How to install GAP
==================

    The file  'install.tex' in 'doc3r2.tar.Z' contains extensive installation
    instructions.  If however, you are one of those  who never  read manuals,
    here is a quick installation guide.

    First for UNIX.

    Make a directory for GAP,  e.g., '~/gap/'  or  '/usr/local/lib/gap/'.

    Unpack the source  archive  'src3r2.tar.Z' into the subdirectory  'src/';
    unpack the library archive  'lib3r2.tar.Z' into the subdirectory  'lib/';
    unpack the documentation    'doc3r2.tar.Z' into the subdirectory  'doc/'.

    If you  have obtained the optional groups and character tables  libraries
    'grp3r2.tar.Z',   'tbl3r2.tar.Z',  'two3r2.tar.Z',   'thr3r2.tar.Z',  and
    'tom3r2.tar.Z',  unpack  them  into the  subdirectories  'grp/',  'tbl/',
    'two/', 'thr/', or 'tom/'.

    Change into 'src/' and execute 'make' to  see a list of possible targets;
    select a target, if in doubt use 'bsd' or 'usg', and make the kernel.

    In an appropriate directory, e.g., '~/bin/' or  '/usr/local/bin/', create
    a shell script that executes the GAP kernel.  This should look like

       exec <gap-directory>/src/gap  -m 4m  -l <gap-directory>/lib/  $*

    The option '-m' specifies the amount  of initial memory; the  option '-l'
    specifies where to find the library, if you get it wrong GAP complains

       gap: hmm, I cannot find 'lib/init.g', maybe use option '-l <libname>'?

    Change into 'doc/' and make the printed manual with the commands

       latex manual;  latex manual;  lp -dvi manual.dvi

    or something similar, according to your local custom for using LaTeX.

    Try something in GAP,  e.g., the following exercises GAP quite a bit

       gap> m11 := Group( (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6) );;
       gap> Number( ConjugacyClasses( m11 ) );

    The result should be 10.

    Next for IBM PC compatibles with an Intel 80386 or 80486 running  MS-DOS.

    Make a directory for GAP, e.g, 'c:\gap\'.

    Put the executable 'gapexe.386' into this directory calling it 'gap.exe'.
    Unpack the library archive   'lib3r2.zoo'  into the subdirectory  'lib\';
    unpack the documentation     'doc3r2.zoo'  into the subdirectory  'doc\'.

    If  you have  obtained the optional  groups and character table libraries
    'grp3r2.zoo', 'tbl3r2.zoo', 'two3r2.zoo', 'thr3r2.zoo', and 'tom3r2.zoo',
    unpack  them into the subdirectories 'grp\', 'tbl\', 'two\',  'thr\', and
    'tom\'.

    In a directory in  your  path,  e.g.,  'c:\bin\',  create  a  batch  file
    'gap.bat' that executes the GAP kernel.  This should look like

       <gap-directory>\gap  -m 4m  -l <gap-directory>\lib\  %1 %2 %3 %4

    The option '-m' specifies the amount  of initial memory; the  option '-l'
    specifies where to find the library, if you get it wrong GAP complains

       gap: hmm, I cannot find 'lib/init.g', maybe use option '-l <libname>'?

    Add the following line to your 'autoexec.bat' file

       SET GO32TMP=<swap-file-directory>

    where <swap-file-directory> should be the directory where you want GAP to
    put the  swap  file,  e.g.,  'c:\tmp'.   The swap  file  will  be  called
    'page????.386' and  is normally removed when GAP exits.  If 'GO32TMP'  is
    not set, 'GCCTMP', 'TMP', 'TEMP' are checked (in this order).  If neither
    is  set, GAP  will not swap to  disk.  *Note  that you must reboot before
    this change in 'autoexec.bat' takes effect*.

    Change into 'doc\' and make the printed manual with the commands

       latex manual;  latex manual;  print manual.dvi

    or something similar, according to your local custom for using LaTeX.

    Try something in GAP,  e.g., the following exercises GAP quite a bit

       gap> m11 := Group( (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6) );;
       gap> Number( ConjugacyClasses( m11 ) );

    The result should be 10.

    Note that GAP for the 386  will  use up  to 128 MByte of extended  memory
    (using XMS, VDISK memory  allocation  strategies)  or up to 128  MByte of
    expanded memory (using VCPI programs, such as QEMM  and 386MAX) and up to
    128 MByte of disk space for swapping.   Further note that GAP for the 386
    will *not* run under Windows (because it does not support DPMI).

    If you hit <ctr>-'C' the DOS  extender ('go32') catches it and aborts GAP
    immediately.  The  keys  <ctr>-'Z' and  <alt>-'C'  can be used instead to
    interupt GAP.

    The arrow keys <left>, <right>, <up>, <down>, <home>, <end>, and <delete>
    can be used for command line editing with their intuitive meaning.

    Pathnames may be  given inside GAP using either shlash ('/') or backslash
    ('\') as a separator (though '\' must be escaped in strings of course).

    The system dependent  part of GAP for the 386 ('sysdos.c') was written by
    Steve Linton (111  Ross  St.,  Cambridge,  CB1 3BS, UK,  +44 223  411661,
    'sl25@cus.cam.ac.uk').  He assignes the copyright to the Lehrstuhl D fuer
    Mathematik.  Many thanks to Steve Linton for his work.

    GAP for the 386 was compiled  with DJ Delorie's port of the Free Software
    Foundation's GNU C compiler version 2.1.  The compiler can be obtained by
    anonymous  'ftp'  from  'grape.ecs.clarkson.edu'  where   it  is  in  the
    directory 'pub/msdos/djgpp'.  Many thanks to the Free Software Foundation
    and DJ Delorie for this amazing piece of work.

    The GNU C compiler is

        Copyright (C) 1989 Free Software Foundation, Inc.
                           675 Mass Ave, Cambridge, MA 02139, USA

    under the  terms  of the GNU General Public License (GPL).  Note that the
    GNU  GPL  states  that the  mere act  of compiling  does not  affect  the
    copyright status of GAP.

    The modifications to the compiler to make it operating under  MS-DOS, the
    functions from  the standard library 'libpc.a', the  modifications of the
    functions from the standard library  'libc.a' to make  them operate under
    MS-DOS, and the DOS extender  'go32' (which is prepended to 'gapexe.386')
    are

        Copyright (C) 1991 DJ Delorie,
                           24 Kirsten Ave, Rochester NH 03867-2954, USA

    also under  the terms of the GNU GPL.  The terms of  the GPL require that
    we make the source code for 'libpc.a' available.  They can be obtained by
    writing to Steve Linton (however, it may be  easier for you to 'ftp' them
    from  'grape.ecs.clarkson.edu'  yourself).   They  also require that  GAP
    falls under the GPL too, i.e., is distributed free   coof users.  We also hope that this number will be  large enough
    to convince various  agencies that GAP is a project worthy of (financial)
    support.   If you publish some result that was partly obtained using GAP,
    we would  appreciate  it  if you  would cite GAP,  just as you would cite
    another paper that you  used.  Again  we  would  appreciate if  you could
    inform us about such a paper.

    We  distribute the  *full  source*  for  everything, the  C code for  the
    kernel, the GAP code for the library, and the LaTeX  code for the manual,
    which has  at present about 800 pages.  So it should be no problem to get
    GAP,  even if you have a rather uncommon system.  Of course, ports to non
    UNIX systems may require some work.   We already  have ports  for  IBM PC
    compatibles with an Intel 80386 or 80486 and for  the Atari ST.   We also
    hope to provide a port  of  GAP 3.2 to the Apple  Macintosh  in the  near
    future (there is already  a port of GAP 3.1).  Note that about 4 MByte of
    main memory and a harddisk are required to run GAP.

    GAP 3.2 can be obtained by anonymous *ftp* from the following servers.

    'samson.math.rwth-aachen.de':
            Lehrstuhl D fur Mathematik, RWTH Aachen, Germany (137.226.152.6).

    'dimacs.rutgers.edu':
            DIMACS, Rutgers, New Brunswick, New Jersey (128.6.75.16).

    'math.ucla.edu':
            Math. Dept., Univ. of California at Los Angeles (128.97.4.254).

    'wuarchive.wustl.edu':
    	    Mathematics Archives, Univ. of Tennessee (128.252.135.4,
            directory '/edu/math/source.code/group.theory/gap').

    'pell.anu.edu.au':
            Math. Research Section, Australian National Univ. (150.203.15.5).

    'ftp' to the server  *closest* to  you, login as user 'ftp' and give your
    full  e-mail  address  as password.  GAP  is in the  directory 'pub/gap'.
    Remember when you transmit  the files  to set the  file  transfer type to
    *binary image*, otherwise you will only receive unusable  garbage.  Those
    servers will always have the latest version of GAP available.

    GAP can also be obtained via *electronic mail*.  To get  one of the files
    mentioned  below  send a message to 'listserv@samson.math.rwth-aachen.de'
    containing a line  'get GAP <file-name>',  e.g., 'get  GAP src3r2.tar.Z'.
    'listserv' will reply by sending you the file as e-mail message.

    Because most files are  large  binary files they will  be  uuencoded  and
    split into  several parts,  each   at  most 64  kBytes  large.   You  can
    concatenate  the parts  by hand,  removing the mail  header, and then use
    'uudecode' to decode them.  We suggest however that you also get 'uud.c',
    which skips  the  mail headers automatically  and  is also able to fix up
    transmission errors caused by 'EBCDIC' machines.  You can also get single
    parts of a file by sending 'get GAP <file-name> <part-nr>'.

    For users in the United Kingdom with only Janet access, neither 'ftp' nor
    the mail server will work (please do *not*  try to  use the mail server).
    Please contact Derek Holt (e-mail address 'dfh@maths.warwick.ac.uk').  He
    has kindly offered us to distribute GAP in the United Kingdom.

    The 'ftp'  directory and  the 'listserv' archive   contain  the following
    files.  Please check first  which files you  need, to  avoid transferring
    those that you don't need.

    'README':               the file you are currently reading.

    GAP version 3 release 2 itself  comes in several files.   You do not need
    all of those files.  All files are 'compress'-ed 'tar' archives.

    'src3r2.tar.Z':         the *source code* for the GAP  kernel.  You  need
                            this unless you get one of the executables below.
                            This file is about 750 KBytes long.

    'lib3r2.tar.Z':         the *library of functions*.  You need this.  This
                            file is about 1000 KBytes long.

    'doc3r2.tar.Z':         the  *documentation*.  Serves  as  LaTeX   source
                            for the printed manual and  online documentation.
                            Contains further installation  information.  This
                            file is about 850 KBytes long.

    'doc3r2.dvi.Z':         the preformatted  documentation.  You  need  this
                            if you do not have a  *big*  TeX.  This  file  is
                            about 1100 KByte long.

    'grp3r2.tar.Z':         various *group libraries*.  Contains for  example
                            all primitive permutation  groups  of  degree  at
                            most 50.  This file is about 50 KByte long.

    'two3r2.tar.Z':         the library of *2-group* of  size  at  most  256.
                            This file is about 650 KByte long.

    'thr3r2.tar.Z':         the library of *3-groups* of  size at  most  729.
                            This file is about 20 KByte long.

    'tbl3r2.tar.Z':         a library of *character tables* including all  of
                            the ATLAS.  This file is about 2050 KByte long.

    'tom3r2.tar.Z':         a library of *table of marks* of various  groups.
                            This file is about 450 KByte long.

    'anupq.tar.Z':          the *ANU PQ* share library package.  This file is
                            about 350 KByte long.

    'nq.tar.Z':             the *NQ*  share  library  package.  This  file is
                            about 100 KByte long.

    'weyl.tar.Z':           the *Weyl* share library package.  This  file  is
                            about 50 KByte long.

    'src3r2.zoo', 'lib3r2.zoo', 'doc3r2.zoo', 'grp3r2.zoo'
    'tbl3r2.zoo', 'two3r2.zoo', 'thr3r2.zoo', 'tom3r2.zoo',
    'anupq.zoo',  'nq.zoo',     'weyl.zoo':
                            'zoo'  archives  containing  *exactly*  the  same
                            files as the 'compress'-ed 'tar' archives  above.
                            The advantage of 'compress'-ed 'tar'  archives is
                            that  'uncompress' and 'tar' are widely available
                            on UNIX systems.  The advantage of 'zoo' archives
                            is  that they are smaller (about  30 percent) and
                            that 'zoo' is more common on PC-s and Atari ST-s.
                            (These files may not be available on all servers)

    We  supply  executables  for  machines  that don't usually  come with a C
    compiler or machines where  the  standard  C  compiler does  not  produce
    optimal results.  If you have one of those machines it will be easier for
    you  to  get  this executable  instead  of compiling  GAP  yourself.  The
    following  executables  are  available  (again  these  files  may not  be
    available on all servers)

    'gapexe.386':           executable for IBM PC compatibles with  an  Intel
                            80386 or  80486 running MS-DOS 5.0 compiled  with
                            the GNU  C  2.2.2  compiler.   See below  for the
                            copyright.  This file is about 500 KByte long.

    'gapexe.next':          executable  for the NeXT (680?0) running NeXTstep
                            3.0  compiled  with  GNU C 2.3.3 compiler.   This
                            file is about 400 KByte long.

    'gapexe.st':            executable  for  Atari  ST  (680?0)  running  TOS
                            compiled with the GNU C compiler.  This  file  is
                            about 450 KByte long.

    'gapexe.su3':           executable for SUN 3 (680?0) running SunOS 4.0 or
                            higher compiled  with the  GNU C  compiler.  This
                            file is about 500 KByte long.

    'gapexe.su4':           executable for SUN 4 (Sparc) running SunOS 4.1 or
                            higher  compiled with the GNU  C  2.3.2 compiler.
                            This file is about 600 KByte long.

    The following support files are also available (and again these files may
    not be available on all servers)

    'compress.tar':         'compress' version 4.1.  You  need  this  program
                            to uncompress  the  compressed  tar  files.  Note
                            however, that almost all UNIX systems  these days
                            already come with an executable 'compress'.  This
                            file is about 90 KByte long.

    'patch.tar.Z':          Larry  Wall's  'patch'  program  version  2.0.2.0
                            (patchlevel 12u4).  This program  can  be used to
                            automatically  apply upgrades.   Note  that older
                            versions  of 'patch' are *not* able to understand
                            the  unified 'diff'  format  used  in the upgrade
                            files.  This file is about 70 KByte long.

    'uud.c':                'uud' version 3.4.  'uud' is much better than the
                            'uudecode' that comes  with  most  UNIX  systems.
                            This file is about 12 KByte long.

    'zoo21.tar.Z':          Rahul Dhesi's  'zoo' archiver  version  2.1.  You
                            need this  to  unpack  the  *zoo-archives*.  Note
                            that the widespread version 2.01 will *not* work.
                            This file is about 250 KByte long.

    'zooexe.386':           Executable of 'zoo' for IBM PC compatibles.  This
                            file is about 55 KByte long.

    'zooexe.st':            Executable of 'zoo' for the Atari ST.  This  file
                            is about 80 KByte long.


How to install GAP
==================

    The file  'install.tex' in 'doc3r2.tar.Z' contains extensive installation
    instructions.  If however, you are one of those  who never  read manuals,
    here is a quick installation guide.

    First for UNIX.

    Make a directory for GAP,  e.g., '~/gap/'  or  '/usr/local/lib/gap/'.

    Unpack the source  archive  'src3r2.tar.Z' into the subdirectory  'src/';
    unpack the library archive  'lib3r2.tar.Z' into the subdirectory  'lib/';
    unpack the documentation    'doc3r2.tar.Z' into the subdirectory  'doc/'.

    If you  have obtained the optional groups and character tables  libraries
    'grp3r2.tar.Z',   'tbl3r2.tar.Z',  'two3r2.tar.Z',   'thr3r2.tar.Z',  and
    'tom3r2.tar.Z',  unpack  them  into the  subdirectories  'grp/',  'tbl/',
    'two/', 'thr/', or 'tom/'.

    Change into 'src/' and execute 'make' to  see a list of possible targets;
    select a target, if in doubt use 'bsd' or 'usg', and make the kernel.

    In an appropriate directory, e.g., '~/bin/' or  '/usr/local/bin/', create
    a shell script that executes the GAP kernel.  This should look like

       exec <gap-directory>/src/gap  -m 4m  -l <gap-directory>/lib/  $*

    The option '-m' specifies the amount  of initial memory; the  option '-l'
    specifies where to find the library, if you get it wrong GAP complains

       gap: hmm, I cannot find 'lib/init.g', maybe use option '-l <libname>'?

    Change into 'doc/' and make the printed manual with the commands

       latex manual;  latex manual;  lp -dvi manual.dvi

    or something similar, according to your local custom for using LaTeX.

    Try something in GAP,  e.g., the following exercises GAP quite a bit

       gap> m11 := Group( (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6) );;
       gap> Number( ConjugacyClasses( m11 ) );

    The result should be 10.

    Next for IBM PC compatibles with an Intel 80386 or 80486 running  MS-DOS.

    Make a directory for GAP, e.g, 'c:\gap\'.

    Put the executable 'gapexe.386' into this directory calling it 'gap.exe'.
    Unpack the library archive   'lib3r2.zoo'  into the subdirectory  'lib\';
    unpack the documentation     'doc3r2.zoo'  into the subdirectory  'doc\'.

    If  you have  obtained the optional  groups and character table libraries
    'grp3r2.zoo', 'tbl3r2.zoo', 'two3r2.zoo', 'thr3r2.zoo', and 'tom3r2.zoo',
    unpack  them into the subdirectories 'grp\', 'tbl\', 'two\',  'thr\', and
    'tom\'.

    In a directory in  your  path,  e.g.,  'c:\bin\',  create  a  batch  file
    'gap.bat' that executes the GAP kernel.  This should look like

       <gap-directory>\gap  -m 4m  -l <gap-directory>\lib\  %1 %2 %3 %4

    The option '-m' specifies the amount  of initial memory; the  option '-l'
    specifies where to find the library, if you get it wrong GAP complains

       gap: hmm, I cannot find 'lib/init.g', maybe use option '-l <libname>'?

    Add the following line to your 'autoexec.bat' file

       SET GO32TMP=<swap-file-directory>

    where <swap-file-directory> should be the directory where you want GAP to
    put the  swap  file,  e.g.,  'c:\tmp'.   The swap  file  will  be  called
    'page????.386' and  is normally removed when GAP exits.  If 'GO32TMP'  is
    not set, 'GCCTMP', 'TMP', 'TEMP' are checked (in this order).  If neither
    is  set, GAP  will not swap to  disk.  *Note  that you must reboot before
    this change in 'autoexec.bat' takes effect*.

    Change into 'doc\' and make the printed manual with the commands

       latex manual;  latex manual;  print manual.dvi

    or something similar, according to your local custom for using LaTeX.

    Try something in GAP,  e.g., the following exercises GAP quite a bit

       gap> m11 := Group( (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6) );;
       gap> Number( ConjugacyClasses( m11 ) );

    The result should be 10.

    Note that GAP for the 386  will  use up  to 128 MByte of extended  memory
    (using XMS, VDISK memory  allocation  strategies)  or up to 128  MByte of
    expanded memory (using VCPI programs, such as QEMM  and 386MAX) and up to
    128 MByte of disk space for swapping.   Further note that GAP for the 386
    will *not* run under Windows (because it does not support DPMI).

    If you hit <ctr>-'C' the DOS  extender ('go32') catches it and aborts GAP
    immediately.  The  keys  <ctr>-'Z' and  <alt>-'C'  can be used instead to
    interupt GAP.

    The arrow keys <left>, <right>, <up>, <down>, <home>, <end>, and <delete>
    can be used for command line editing with their intuitive meaning.

    Pathnames may be  given inside GAP using either shlash ('/') or backslash
    ('\') as a separator (though '\' must be escaped in strings of course).

    The system dependent  part of GAP for the 386 ('sysdos.c') was written by
    Steve Linton (111  Ross  St.,  Cambridge,  CB1 3BS, UK,  +44 223  411661,
    'sl25@cus.cam.ac.uk').  He assignes the copyright to the Lehrstuhl D fuer
    Mathematik.  Many thanks to Steve Linton for his work.

    GAP for the 386 was compiled  with DJ Delorie's port of the Free Software
    Foundation's GNU C compiler version 2.1.  The compiler can be obtained by
    anonymous  'ftp'  from  'grape.ecs.clarkson.edu'  where   it  is  in  the
    directory 'pub/msdos/djgpp'.  Many thanks to the Free Software Foundation
    and DJ Delorie for this amazing piece of work.

    The GNU C compiler is

        Copyright (C) 1989 Free Software Foundation, Inc.
                           675 Mass Ave, Cambridge, MA 02139, USA

    under the  terms  of the GNU General Public License (GPL).  Note that the
    GNU  GPL  states  that the  mere act  of compiling  does not  affect  the
    copyright status of GAP.

    The modifications to the compiler to make it operating under  MS-DOS, the
    functions from  the standard library 'libpc.a', the  modifications of the
    functions from the standard library  'libc.a' to make  them operate under
    MS-DOS, and the DOS extender  'go32' (which is prepended to 'gapexe.386')
    are

        Copyright (C) 1991 DJ Delorie,
                           24 Kirsten Ave, Rochester NH 03867-2954, USA

    also under  the terms of the GNU GPL.  The terms of  the GPL require that
    we make the source code for 'libpc.a' available.  They can be obtained by
    writing to Steve Linton (however, it may be  easier for you to 'ftp' them
    from  'grape.ecs.clarkson.edu'  yourself).   They  also require that  GAP
    falls under the GPL too, i.e., is distributed free   coof users.  We also hope that this number will be  large enough
    to convince various  agencies that GAP is a project worthy of (financial)
    support.   If you publish some result that was partly obtained using GAP,
    we would  appreciate  it  if you  would cite GAP,  just as you would cite
    another paper that you  used.  Again  we  would  appreciate if  you could
    inform us about such a paper.

    We  distribute the  *full  source*  for  everything, the  C code for  the
    kernel, the GAP code for the library, and the LaTeX  code for the manual,
    which has  at present about 800 pages.  So it should be no problem to get
    GAP,  even if you have a rather uncommon system.  Of course, ports to non
    UNIX systems may require some work.   We already  have ports  for  IBM PC
    compatibles with an Intel 80386 or 80486 and for  the Atari ST.   We also
    hope to provide a port  of  GAP 3.2 to the Apple  Macintosh  in the  near
    future (there is already  a port of GAP 3.1).  Note that about 4 MByte of
    main memory and a harddisk are required to run GAP.

    GAP 3.2 can be obtained by anonymous *ftp* from the following servers.

    'samson.math.rwth-aachen.de':
            Lehrstuhl D fur Mathematik, RWTH Aachen, Germany (137.226.152.6).

    'dimacs.rutgers.edu':
            DIMACS, Rutgers, New Brunswick, New Jersey (128.6.75.16).

    'math.ucla.edu':
            Math. Dept., Univ. of California at Los Angeles (128.97.4.254).

    'wuarchive.wustl.edu':
    	    Mathematics Archives, Univ. of Tennessee (128.252.135.4,
            directory '/edu/math/source.code/group.theory/gap').

    'pell.anu.edu.au':
            Math. Research Section, Australian National Univ. (150.203.15.5).

    'ftp' to the server  *closest* to  you, login as user 'ftp' and give your
    full  e-mail  address  as password.  GAP  is in the  directory 'pub/gap'.
    Remember when you transmit  the files  to set the  file  transfer type to
    *binary image*, otherwise you will only receive unusable  garbage.  Those
    servers will always have the latest version of GAP available.

    GAP can also be obtained via *electronic mail*.  To get  one of the files
    mentioned  below  send a message to 'listserv@samson.math.rwth-aachen.de'
    containing a line  'get GAP <file-name>',  e.g., 'get  GAP src3r2.tar.Z'.
    'listserv' will reply by sending you the file as e-mail message.

    Because most files are  large  binary files they will  be  uuencoded  and
    split into  several parts,  each   at  most 64  kBytes  large.   You  can
    concatenate  the parts  by hand,  removing the mail  header, and then use
    'uudecode' to decode them.  We suggest however that you also get 'uud.c',
    which skips  the  mail headers automatically  and  is also able to fix up
    transmission errors caused by 'EBCDIC' machines.  You can also get single
    parts of a file by sending 'get GAP <file-name> <part-nr>'.

    For users in the United Kingdom with only Janet access, neither 'ftp' nor
    the mail server will work (please do *not*  try to  use the mail server).
    Please contact Derek Holt (e-mail address 'dfh@maths.warwick.ac.uk').  He
    has kindly offered us to distribute GAP in the United Kingdom.

    The 'ftp'  directory and  the 'listserv' archive   contain  the following
    files.  Please check first  which files you  need, to  avoid transferring
    those that you don't need.

    'README':               the file you are currently reading.

    GAP version 3 release 2 itself  comes in several files.   You do not need
    all of those files.  All files are 'compress'-ed 'tar' archives.

    'src3r2.tar.Z':         the *source code* for the GAP  kernel.  You  need
                            this unless you get one of the executables below.
                            This file is about 750 KBytes long.

    'lib3r2.tar.Z':         the *library of functions*.  You need this.  This
                            file is about 1000 KBytes long.

    'doc3r2.tar.Z':         the  *documentation*.  Serves  as  LaTeX   source
                            for the printed manual and  online documentation.
                            Contains further installation  information.  This
                            file is about 850 KBytes long.

    'doc3r2.dvi.Z':         the preformatted  documentation.  You  need  this
                            if you do not have a  *big*  TeX.  This  file  is
                            about 1100 KByte long.

    'grp3r2.tar.Z':         various *group libraries*.  Contains for  example
                            all primitive permutation  groups  of  degree  at
                            most 50.  This file is about 50 KByte long.

    'two3r2.tar.Z':         the library of *2-group* of  size  at  most  256.
                            This file is about 650 KByte long.

    'thr3r2.tar.Z':         the library of *3-groups* of  size at  most  729.
                            This file is about 20 KByte long.

    'tbl3r2.tar.Z':         a library of *character tables* including all  of
                            the ATLAS.  This file is about 2050 KByte long.

    'tom3r2.tar.Z':         a library of *table of marks* of various  groups.
                            This file is about 450 KByte long.

    'anupq.tar.Z':          the *ANU PQ* share library package.  This file is
                            about 350 KByte long.

    'nq.tar.Z':             the *NQ*  share  library  package.  This  file is
                            about 100 KByte long.

    'weyl.tar.Z':           the *Weyl* share library package.  This  file  is
                            about 50 KByte long.

    'src3r2.zoo', 'lib3r2.zoo', 'doc3r2.zoo', 'grp3r2.zoo'
    'tbl3r2.zoo', 'two3r2.zoo', 'thr3r2.zoo', 'tom3r2.zoo',
    'anupq.zoo',  'nq.zoo',     'weyl.zoo':
                            'zoo'  archives  containing  *exactly*  the  same
                            files as the 'compress'-ed 'tar' archives  above.
                            The advantage of 'compress'-ed 'tar'  archives is
                            that  'uncompress' and 'tar' are widely available
                            on UNIX systems.  The advantage of 'zoo' archives
                            is  that they are smaller (about  30 percent) and
                            that 'zoo' is more common on PC-s and Atari ST-s.
                            (These files may not be available on all servers)

    We  supply  executables  for  machines  that don't usually  come with a C
    compiler or machines where  the  standard  C  compiler does  not  produce
    optimal results.  If you have one of those machines it will be easier for
    you  to  get  this executable  instead  of compiling  GAP  yourself.  The
    following  executables  are  available  (again  these  files  may not  be
    available on all servers)

    'gapexe.386':           executable for IBM PC compatibles with  an  Intel
                            80386 or  80486 running MS-DOS 5.0 compiled  with
                            the GNU  C  2.2.2  compiler.   See below  for the
                            copyright.  This file is about 500 KByte long.

    'gapexe.next':          executable  for the NeXT (680?0) running NeXTstep
                            3.0  compiled  with  GNU C 2.3.3 compiler.   This
                            file is about 400 KByte long.

    'gapexe.st':            executable  for  Atari  ST  (680?0)  running  TOS
                            compiled with the GNU C compiler.  This  file  is
                            about 450 KByte long.

    'gapexe.su3':           executable for SUN 3 (680?0) running SunOS 4.0 or
                            higher compiled  with the  GNU C  compiler.  This
                            file is about 500 KByte long.

    'gapexe.su4':           executable for SUN 4 (Sparc) running SunOS 4.1 or
                            higher  compiled with the GNU  C  2.3.2 compiler.
                            This file is about 600 KByte long.

    The following support files are also available (and again these files may
    not be available on all servers)

    'compress.tar':         'compress' version 4.1.  You  need  this  program
                            to uncompress  the  compressed  tar  files.  Note
                            however, that almost all UNIX systems  these days
                            already come with an executable 'compress'.  This
                            file is about 90 KByte long.

    'patch.tar.Z':          Larry  Wall's  'patch'  program  version  2.0.2.0
                            (patchlevel 12u4).  This program  can  be used to
                            automatically  apply upgrades.   Note  that older
                            versions  of 'patch' are *not* able to understand
                            the  unified 'diff'  format  used  in the upgrade
                            files.  This file is about 70 KByte long.

    'uud.c':                'uud' version 3.4.  'uud' is much better than the
                            'uudecode' that comes  with  most  UNIX  systems.
                            This file is about 12 KByte long.

    'zoo21.tar.Z':          Rahul Dhesi's  'zoo' archiver  version  2.1.  You
                            need this  to  unpack  the  *zoo-archives*.  Note
                            that the widespread version 2.01 will *not* work.
                            This file is about 250 KByte long.

    'zooexe.386':           Executable of 'zoo' for IBM PC compatibles.  This
                            file is about 55 KByte long.

    'zooexe.st':            Executable of 'zoo' for the Atari ST.  This  file
                            is about 80 KByte long.


How to install GAP
==================

    The file  'install.tex' in 'doc3r2.tar.Z' contains extensive installation
    instructions.  If however, you are one of those  who never  read manuals,
    here is a quick installation guide.

    First for UNIX.

    Make a directory for GAP,  e.g., '~/gap/'  or  '/usr/local/lib/gap/'.

    Unpack the source  archive  'src3r2.tar.Z' into the subdirectory  'src/';
    unpack the library archive  'lib3r2.tar.Z' into the subdirectory  'lib/';
    unpack the documentation    'doc3r2.tar.Z' into the subdirectory  'doc/'.

    If you  have obtained the optional groups and character tables  libraries
    'grp3r2.tar.Z',   'tbl3r2.tar.Z',  'two3r2.tar.Z',   'thr3r2.tar.Z',  and
    'tom3r2.tar.Z',  unpack  them  into the  subdirectories  'grp/',  'tbl/',
    'two/', 'thr/', or 'tom/'.

    Change into 'src/' and execute 'make' to  see a list of possible targets;
    select a target, if in doubt use 'bsd' or 'usg', and make the kernel.

    In an appropriate directory, e.g., '~/bin/' or  '/usr/local/bin/', create
    a shell script that executes the GAP kernel.  This should look like

       exec <gap-directory>/src/gap  -m 4m  -l <gap-directory>/lib/  $*

    The option '-m' specifies the amount  of initial memory; the  option '-l'
    specifies where to find the library, if you get it wrong GAP complains

       gap: hmm, I cannot find 'lib/init.g', maybe use option '-l <libname>'?

    Change into 'doc/' and make the printed manual with the commands

       latex manual;  latex manual;  lp -dvi manual.dvi

    or something similar, according to your local custom for using LaTeX.

    Try something in GAP,  e.g., the following exercises GAP quite a bit

       gap> m11 := Group( (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6) );;
       gap> Number( ConjugacyClasses( m11 ) );

    The result should be 10.

    Next for IBM PC compatibles with an Intel 80386 or 80486 running  MS-DOS.

    Make a directory for GAP, e.g, 'c:\gap\'.

    Put the executable 'gapexe.386' into this directory calling it 'gap.exe'.
    Unpack the library archive   'lib3r2.zoo'  into the subdirectory  'lib\';
    unpack the documentation     'doc3r2.zoo'  into the subdirectory  'doc\'.

    If  you have  obtained the optional  groups and character table libraries
    'grp3r2.zoo', 'tbl3r2.zoo', 'two3r2.zoo', 'thr3r2.zoo', and 'tom3r2.zoo',
    unpack  them into the subdirectories 'grp\', 'tbl\', 'two\',  'thr\', and
    'tom\'.

    In a directory in  your  path,  e.g.,  'c:\bin\',  create  a  batch  file
    'gap.bat' that executes the GAP kernel.  This should look like

       <gap-directory>\gap  -m 4m  -l <gap-directory>\lib\  %1 %2 %3 %4

    The option '-m' specifies the amount  of initial memory; the  option '-l'
    specifies where to find the library, if you get it wrong GAP complains

       gap: hmm, I cannot find 'lib/init.g', maybe use option '-l <libname>'?

    Add the following line to your 'autoexec.bat' file

       SET GO32TMP=<swap-file-directory>

    where <swap-file-directory> should be the directory where you want GAP to
    put the  swap  file,  e.g.,  'c:\tmp'.   The swap  file  will  be  called
    'page????.386' and  is normally removed when GAP exits.  If 'GO32TMP'  is
    not set, 'GCCTMP', 'TMP', 'TEMP' are checked (in this order).  If neither
    is  set, GAP  will not swap to  disk.  *Note  that you must reboot before
    this change in 'autoexec.bat' takes effect*.

    Change into 'doc\' and make the printed manual with the commands

       latex manual;  latex manual;  print manual.dvi

    or something similar, according to your local custom for using LaTeX.

    Try something in GAP,  e.g., the following exercises GAP quite a bit

       gap> m11 := Group( (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6) );;
       gap> Number( ConjugacyClasses( m11 ) );

    The result should be 10.

    Note that GAP for the 386  will  use up  to 128 MByte of extended  memory
    (using XMS, VDISK memory  allocation  strategies)  or up to 128  MByte of
    expanded memory (using VCPI programs, such as QEMM  and 386MAX) and up to
    128 MByte of disk space for swapping.   Further note that GAP for the 386
    will *not* run under Windows (because it does not support DPMI).

    If you hit <ctr>-'C' the DOS  extender ('go32') catches it and aborts GAP
    immediately.  The  keys  <ctr>-'Z' and  <alt>-'C'  can be used instead to
    interupt GAP.

    The arrow keys <left>, <right>, <up>, <down>, <home>, <end>, and <delete>
    can be used for command line editing with their intuitive meaning.

    Pathnames may be  given inside GAP using either shlash ('/') or backslash
    ('\') as a separator (though '\' must be escaped in strings of course).

    The system dependent  part of GAP for the 386 ('sysdos.c') was written by
    Steve Linton (111  Ross  St.,  Cambridge,  CB1 3BS, UK,  +44 223  411661,
    'sl25@cus.cam.ac.uk').  He assignes the copyright to the Lehrstuhl D fuer
    Mathematik.  Many thanks to Steve Linton for his work.

    GAP for the 386 was compiled  with DJ Delorie's port of the Free Software
    Foundation's GNU C compiler version 2.1.  The compiler can be obtained by
    anonymous  'ftp'  from  'grape.ecs.clarkson.edu'  where   it  is  in  the
    directory 'pub/msdos/djgpp'.  Many thanks to the Free Software Foundation
    and DJ Delorie for this amazing piece of work.

    The GNU C compiler is

        Copyright (C) 1989 Free Software Foundation, Inc.
                           675 Mass Ave, Cambridge, MA 02139, USA

    under the  terms  of the GNU General Public License (GPL).  Note that the
    GNU  GPL  states  that the  mere act  of compiling  does not  affect  the
    copyright status of GAP.

    The modifications to the compiler to make it operating under  MS-DOS, the
    functions from  the standard library 'libpc.a', the  modifications of the
    functions from the standard library  'libc.a' to make  them operate under
    MS-DOS, and the DOS extender  'go32' (which is prepended to 'gapexe.386')
    are

        Copyright (C) 1991 DJ Delorie,
                           24 Kirsten Ave, Rochester NH 03867-2954, USA

    also under  the terms of the GNU GPL.  The terms of  the GPL require that
    we make the source code for 'libpc.a' available.  They can be obtained by
    writing to Steve Linton (however, it may be  easier for you to 'ftp' them
    from  'grape.ecs.clarkson.edu'  yourself).   They  also require that  GAP
    falls under the GPL too, i.e., is distributed free   coof users.  We also hope that this number will be  large enough
    to convince various  agencies that GAP is a project worthy of (financial)
    support.   If you publish some result that was partly obtained using GAP,
    we would  appreciate  it  if you  would cite GAP,  just as you would cite
    another paper that you  used.  Again  we  would  appreciate if  you could
    inform us about such a paper.

    We  distribute the  *full  source*  for  everything, the  C code for  the
    kernel, the GAP code for the library, and the LaTeX  code for the manual,
    which has  at present about 800 pages.  So it should be no problem to get
    GAP,  even if you have a rather uncommon system.  Of course, ports to non
    UNIX systems may require some work.   We already  have ports  for  IBM PC
    compatibles with an Intel 80386 or 80486 and for  the Atari ST.   We also
    hope to provide a port  of  GAP 3.2 to the Apple  Macintosh  in the  near
    future (there is already  a port of GAP 3.1).  Note that about 4 MByte of
    main memory and a harddisk are required to run GAP.

    GAP 3.2 can be obtained by anonymous *ftp* from the following servers.

    'samson.math.rwth-aachen.de':
            Lehrstuhl D fur Mathematik, RWTH Aachen, Germany (137.226.152.6).

    'dimacs.rutgers.edu':
            DIMACS, Rutgers, New Brunswick, New Jersey (128.6.75.16).

    'math.ucla.edu':
            Math. Dept., Univ. of California at Los Angeles (128.97.4.254).

    'wuarchive.wustl.edu':
    	    Mathematics Archives, Univ. of Tennessee (128.252.135.4,
            directory '/edu/math/source.code/group.theory/gap').

    'pell.anu.edu.au':
            Math. Research Section, Australian National Univ. (150.203.15.5).

    'ftp' to the server  *closest* to  you, login as user 'ftp' and give your
    full  e-mail  address  as password.  GAP  is in the  directory 'pub/gap'.
    Remember when you transmit  the files  to set the  file  transfer type to
    *binary image*, otherwise you will only receive unusable  garbage.  Those
    servers will always have the latest version of GAP available.

    GAP can also be obtained via *electronic mail*.  To get  one of the files
    mentioned  below  send a message to 'listserv@samson.math.rwth-aachen.de'
    containing a line  'get GAP <file-name>',  e.g., 'get  GAP src3r2.tar.Z'.
    'listserv' will reply by sending you the file as e-mail message.

    Because most files are  large  binary files they will  be  uuencoded  and
    split into  several parts,  each   at  most 64  kBytes  large.   You  can
    concatenate  the parts  by hand,  removing the mail  header, and then use
    'uudecode' to decode them.  We suggest however that you also get 'uud.c',
    which skips  the  mail headers automatically  and  is also able to fix up
    transmission errors caused by 'EBCDIC' machines.  You can also get single
    parts of a file by sending 'get GAP <file-name> <part-nr>'.

    For users in the United Kingdom with only Janet access, neither 'ftp' nor
    the mail server will work (please do *not*  try to  use the mail server).
    Please contact Derek Holt (e-mail address 'dfh@maths.warwick.ac.uk').  He
    has kindly offered us to distribute GAP in the United Kingdom.

    The 'ftp'  directory and  the 'listserv' archive   contain  the following
    files.  Please check first  which files you  need, to  avoid transferring
    those that you don't need.

    'README':               the file you are currently reading.

    GAP version 3 release 2 itself  comes in several files.   You do not need
    all of those files.  All files are 'compress'-ed 'tar' archives.

    'src3r2.tar.Z':         the *source code* for the GAP  kernel.  You  need
                            this unless you get one of the executables below.
                            This file is about 750 KBytes long.

    'lib3r2.tar.Z':         the *library of functions*.  You need this.  This
                            file is about 1000 KBytes long.

    'doc3r2.tar.Z':         the  *documentation*.  Serves  as  LaTeX   source
                            for the printed manual and  online documentation.
                            Contains further installation  information.  This
                            file is about 850 KBytes long.

    'doc3r2.dvi.Z':         the preformatted  documentation.  You  need  this
                            if you do not have a  *big*  TeX.  This  file  is
                            about 1100 KByte long.

    'grp3r2.tar.Z':         various *group libraries*.  Contains for  example
                            all primitive permutation  groups  of  degree  at
                            most 50.  This file is about 50 KByte long.

    'two3r2.tar.Z':         the library of *2-group* of  size  at  most  256.
                            This file is about 650 KByte long.

    'thr3r2.tar.Z':         the library of *3-groups* of  size at  most  729.
                            This file is about 20 KByte long.

    'tbl3r2.tar.Z':         a library of *character tables* including all  of
                            the ATLAS.  This file is about 2050 KByte long.

    'tom3r2.tar.Z':         a library of *table of marks* of various  groups.
                            This file is about 450 KByte long.

    'anupq.tar.Z':          the *ANU PQ* share library package.  This file is
                            about 350 KByte long.

    'nq.tar.Z':             the *NQ*  share  library  package.  This  file is
                            about 100 KByte long.

    'weyl.tar.Z':           the *Weyl* share library package.  This  file  is
                            about 50 KByte long.

    'src3r2.zoo', 'lib3r2.zoo', 'doc3r2.zoo', 'grp3r2.zoo'
    'tbl3r2.zoo', 'two3r2.zoo', 'thr3r2.zoo', 'tom3r2.zoo',
    'anupq.zoo',  'nq.zoo',     'weyl.zoo':
                            'zoo'  archives  containing  *exactly*  the  same
                            files as the 'compress'-ed 'tar' archives  above.
                            The advantage of 'compress'-ed 'tar'  archives is
                            that  'uncompress' and 'tar' are widely available
                            on UNIX systems.  The advantage of 'zoo' archives
                            is  that they are smaller (about  30 percent) and
                            that 'zoo' is more common on PC-s and Atari ST-s.
                            (These files may not be available on all servers)

    We  supply  executables  for  machines  that don't usually  come with a C
    compiler or machines where  the  standard  C  compiler does  not  produce
    optimal results.  If you have one of those machines it will be easier for
    you  to  get  this executable  instead  of compiling  GAP  yourself.  The
    following  executables  are  available  (again  these  files  may not  be
    available on all servers)

    'gapexe.386':           executable for IBM PC compatibles with  an  Intel
                            80386 or  80486 running MS-DOS 5.0 compiled  with
                            the GNU  C  2.2.2  compiler.   See below  for the
                            copyright.  This file is about 500 KByte long.

    'gapexe.next':          executable  for the NeXT (680?0) running NeXTstep
                            3.0  compiled  with  GNU C 2.3.3 compiler.   This
                            file is about 400 KByte long.

    'gapexe.st':            executable  for  Atari  ST  (680?0)  running  TOS
                            compiled with the GNU C compiler.  This  file  is
                            about 450 KByte long.

    'gapexe.su3':           executable for SUN 3 (680?0) running SunOS 4.0 or
                            higher compiled  wi