> < ^ Date: Fri, 19 Feb 1993 20:11:14 +0100
> < ^ From: Frank Celler <frank.celler@math.rwth-aachen.de >
^ 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



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.

                                                            Il est trop tard,
                                                  il sera toujours trop tard.
                                                         (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 );

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

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 );

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 );

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 ) );

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 );
gap> Size( Intersection( cmpl1, Kernel( blockhom1 ) ) );
gap> Closure( cmpl1, Kernel( blockhom1 ) ) = cube1;

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

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

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

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

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 );
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 );
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 ) );
gap> cmpl2 := Stabilizer(cube2,[1,2,3,4,5,6,7,9,10,12,14,16],OnSets);;
gap> IsIsomorphism( OperationHomomorphism( cmpl2, cube2b ) );

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

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 );
gap> Size( cube1 ) * Size( cube2 );

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

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

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

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

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

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.

Lehrstuhl D fur Mathematik, RWTH Aachen, Germany (

DIMACS, Rutgers, New Brunswick, New Jersey (

        Math. Dept., Univ. of California at Los Angeles (

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

Math. Research Section, Australian National Univ. (

'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
                        (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

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')

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 freely, which it basically
does anyhow.

The functions in 'libc.a' that GAP for the 386 uses are

Copyright (c) 1988 Regents of the University of California.

under the following terms

All rights reserved.

Redistribution and use in source and binary forms are permitted
provided that the above copyright notice and this paragraph are
duplicated in all such forms and that any documentation, advertising
materials, and other materials related to such distribution and use
acknowledge that the software was developed by the University of
California, Berkeley. The name of the University may not be used to
endorse or promote products derived from this software without
specific prior written permission.


The Future of GAP
                                                 See ye not all these things?
                                                       Verily I say unto you,
                                                 there shall not be left here
                                                      one stone upon another,
                                               that shall not be thrown down.
                                                               (Matthew 24:2)

Clearly GAP will contain bugs, as any system of this size, though
currently we know none. Also there are things that we feel are still
missing, and that we would like to include into GAP. We will continue to
improve and extend GAP. We will release new versions quite regulary now,
and about three or four upgrades a year are planned. Make sure to get
these, since they will in particular contain bug-fixes.

We are committed however, to staying upward compatible from now on in
future releases. That means that everything that works now will also
work in those future releases. This is different from the quite radical
step from GAP 2.4 to GAP 3.1, in which almost everything was changed.

Of course, we have ideas about what we want to have in future versions of
GAP. However we are also looking forward to your comments or

The GAP Forum

We have also established a GAP forum, where interested users can discuss
GAP related topics by e-mail. In particular this forum is for questions
about GAP, general comments, bug reports, and maybe bug fixes. We, the
developers of GAP, will read this forum and answer questions and
comments, and distribute bug fixes. Of course others are also invited to
answer questions, etc. We will also announce future releases of GAP on
this forum. So in order to be informed about bugs and their fixes as
well as about additions to GAP we recommend that you subscribe to the GAP

To subscribe send a message to 'listserv@samson.math.rwth-aachen.de'
containing the line 'subscribe gap-forum <your-name>', where <your-name>
should be your full name, not your e-mail address. You will receive an
acknowledgement, and from then on all e-mail messages sent to

'listserv@samson.math.rwth-aachen.de' also accepts the following
requests: 'help' for a short help on how to use 'listserv', 'unsubscribe
gap-forum' to unsubscribe again, 'recipients gap-forum' to get a list of
subscribers, and 'statistics gap-forum' to see how many e-mail messages
each subscriber has sent so far.

If you have further questions or comments do not hesitate to write to me

Thank you for your attention, 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

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