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| Title: | Mathematics at DEC |
|
| Moderator: | RUSURE::EDP |
|
| Created: | Mon Feb 03 1986 |
| Last Modified: | Fri Jun 06 1997 |
| Last Successful Update: | Fri Jun 06 1997 |
| Number of topics: | 2083 |
| Total number of notes: | 14613 |
1882.0. "Groups, Algorithms and Programming" by STAR::PRAETORIUS (I have faith in questioning) Thu Jul 21 1994 20:18
[I have neither the time nor the knowledge to mess with this, but thought
it might be interesting to somebody out there. RMP]
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 3, GAP 3.3 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 3.3" tells you about the new features in GAP
3.3. The section "Copyright" states the terms under which you can copy
GAP. 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 3 #
## # 07 Nov 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 3, or GAP 3.3 for short. This is the second
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 solvable
irreducible matrix groups from Mark Short, 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 3.3 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 six 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 nilpotent quotients of finitely presented groups 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. The
*Grape* package, written by L. Soicher, consists of functions to work
with graphs, and contains also functions to call the program 'naughty',
written by B. McKay, which computes automorphism groups of graphs. The
*XGAP* package, written by F. Celler and S. Keitemeier, allows the user
to run GAP under the X window system and e.g. work with subgroup lattices
interactively. The *Sisyphos* package, written by M. Wursthorn, consists
of a program that computes automorphisms of <p>-groups and functions to
to interface this program with GAP.
What's New in 3.3
=================
GAP 3.3 is mainly a bug fix release, in which we have tried to fix all
the problems found in the past.
GAP also runs on more systems than ever before. New are the support for
IBM PC with OS/2, Apple Macintosh with Mac Programmers Workshop, and DEC
VAX with VMS.
GAP 3.3 contains a new implementation of the function that computes the
subgroup lattice of a reasonably small group.
The library of solvable irreducible matrix groups from Mark Short is new.
The Grape, XGAP, and Sisyphos packages are also new.
Copyright
=========
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
Copyright (C) 1993 by Lehrstuhl D f"ur Mathematik
RWTH, Templergraben 64, D 52056 Aachen, Germany
GAP can be copied and distributed freely for any non-commercial purpose.
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 copy GAP for somebody else, you may ask this person for refund of
your expenses. This should cover cost of media, copying and shipping.
You are not allowed to ask for more than this. In any case you must give
a copy of this copyright notice along with the program.
If you obtain GAP please send us a short notice to that effect, e.g., an
e-mail message to the address '[email protected]',
containing your full name and address. This allows us to keep track of
the number of GAP users.
If you publish a mathematical result that was partly obtained using GAP,
please cite GAP, just as you would cite another paper that you used.
Also we would appreciate it if you could inform us about such a paper.
You are permitted to modify and redistribute GAP, but you are not allowed
to restrict further redistribution. That is to say proprietary
modifications will not be allowed. We want all versions of GAP to remain
free.
If you modify any part of GAP and redistribute it, you must supply a
'README' document. This should specify what modifications you made in
which files. We do not want to take credit or be blamed for your
modifications.
Of course we are interested in all of your modifications. In particular
we would like to see bug-fixes, improvements and new functions. So again
we would appreciate it if you would inform us about all modifications you
make.
GAP is distributed by us without any warranty, to the extent permitted by
applicable state law. We distribute GAP *as is* without warranty of any
kind, either expressed or implied, including, but not limited to, the
implied warranties of merchantability and fitness for a particular
purpose.
The entire risk as to the quality and performance of the program is with
you. Should GAP prove defective, you assume the cost of all necessary
servicing, repair or correction.
In no case unless required by applicable law will we, and/or any other
party who may modify and redistribute GAP as permitted above, be liable
to you for damages, including lost profits, lost monies or other special,
incidental or consequential damages arising out of the use or inability
to use GAP.
The system dependent part of GAP for IBM PCs with MS-DOS was written by
Steve Linton (111 Ross St., Cambridge, CB1 3BS, UK, +44 223 411661,
'[email protected]'). The system dependent part of GAP for IBM PCs with
OS/2 was written by Harald Boegeholz ('[email protected]').
The system dependent part of GAP for VAX machines with VMS was written by
Paul Doyle ('[email protected]'). The system dependent part of GAP
for the Macintosh with Mac Programmers Workshop (MPW) was written by Dave
Bayer ('[email protected]'). Many thanks to them for assigning
the copyright to Lehrstuhl D f"ur Mathematik and/or allowing us to
distribute their code.
Some of the executables come with additional copyright notices. Please
check the files that come with the executables.
How to get GAP
==============
GAP is distributed *free of charge*. You can obtain it via 'ftp' and
give it away to your colleagues.
If you get GAP, we would appreciate it if you could notify us, e.g., by
sending a short e-mail message to '[email protected]',
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 900 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 under MS-DOS, Windows, or OS/2,
for the Apple Macintosh under MPW (we hope to provide a standalone port
soon), for the Atari ST under TOS, and for DEC VAX and AXP under VMS.
Note that about 4 MByte of main memory and a harddisk are required to run
GAP.
GAP 3.3 (currently at patchlevel 0) 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);
directory '/pub/gap/'.
'dimacs.rutgers.edu':
DIMACS, Rutgers, New Brunswick, New Jersey (128.6.75.16);
directory '/pub/gap/'.
'math.ucla.edu':
Math. Dept., Univ. of California at Los Angeles (128.97.4.254);
directory '/pub/gap/'.
'wuarchive.wustl.edu':
Math. Archives, Washington Univ. at St. Louis (128.252.135.4);
directory '/edu/math/source.code/group.theory/gap/'.
'dehn.mth.pdx.edu':
PSU Mathematics Department, Portland State Univ (131.252.40.89);
directory '/mirror/gap/'.
'pell.anu.edu.au':
Math. Research Section, Australian National Univ. (150.203.15.5);
directory '/pub/gap/'.
'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.
For users in the United Kingdom with Janet access 'ftp' may not work.
Please contact Derek Holt (e-mail address '[email protected]'). He
has kindly offered us to distribute GAP in the United Kingdom.
The 'ftp' directory contains 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.
'gap3r3p0.zoo': This file contains the *complete* distribution
of GAP version 3 release 3 current patchlevel 0.
It is a 'zoo' archive approximately 7 MByte big.
'unzoo.c': A simple 'zoo' archive extractor, which should be
used to unpack the distribution. The 'utils'
subdirectory contains ready compiled executables
for common systems.
More files are in the following *subdirectories*:
'bin': This directory contains *executables* for systems
that dont come with a C compiler or where another
C compiler produces a faster executable. The
'KERNELS' file tells you which executables are
here.
'split': This directory contains the complete distribution
of GAP 3r3p0 in several 'zoo' archives. This
allows you to get only the parts that you are
really interested in. The 'SPLIT' file tells you
which archive contains what.
'utils': This directory contains several utilities that
you may need to get or upgrade GAP, e.g., 'unzoo'
and 'patch'. The 'UTILS' file tells you which
files are here.
How to install GAP
==================
The file 'doc/install.tex' contains extensive installation instructions.
If however, you are one of those who never read manuals, here is a quick
installation guide.
Ho to install GAP on an IBM PC, Apple Mac, Atari ST, DEC VAX, or DEC AXP
------------------------------------------------------------------------
First go to a directory where you want to install GAP , e.g., 'c:\'.
GAP will be installed in a subdirectory 'gap3r3p0\' of this directory.
You can later move GAP to another location, for example you can first
install it in 'd:\tmp\' and once it works move it to 'c:\'.
Get the GAP distribution onto your computer. One usual way would be to
get the distribution with 'ftp' onto some UNIX workstation and to
download it from there onto your computer, for example with 'kermit'.
Remember that the distribution consists of binary files and that you must
transmit them in binary mode. Another possibility is that you got a set
of floppy disks.
In any case you need the distribution 'gap3r3p0.zoo', the appropriate zoo
archive extractor, e.g., 'unzoo-ibm-i386-msdos.exe', which is in the
subdirectory 'util/' on the 'ftp' server, and the appropriate executable,
e.g., 'bin3r3p0-ibm-i386-msdos.zoo', which is in the subdirectory 'bin/'.
You may have to get the latter 2 files from 'samson.math.rwth-aachen.de',
because some 'ftp' servers may not keep it. We recommend that you use
'unzoo' even if you already have 'zoo' on your system, because 'unzoo'
automatically translates text files to the appropriate local format.
Unpack the executable and the distribution with the command
unzoo -x -o bin3r3p0.zoo
unzoo -x -o gap3r3p0.zoo
Now change into the 'gap3r3p0\bin\' subdirectory and follow the
instructions in the appropriate readme file, e.g., 'README.DOS'.
How to install GAP on a UNIX computer
-------------------------------------
First go to a directory where you want to install GAP, e.g., your home
directory or '/usr/local/lib/'. GAP will be installed in a subdirectory
called 'gap3r3p0/' of this directory. You can later move GAP to another
location, for example you can first install it in your home directory and
once it works move it to '/usr/local/lib/'.
Get the distribution 'gap3r3p0.zoo' and the source for the zoo archive
extractor 'unzoo.c'. How you can get those files is described in the
section "How to get GAP". The usual way would be to get it with
'ftp'onto your machine. Remember that the distribution consists of
binary files and that you must transmit them in binary mode.
Compile the zoo archive extractor 'unzoo' with the command
cc -o unzoo -DSYS_IS_UNIX -O unzoo.c
Unpack the distribution with the command
unzoo -x -o gap3r3p0.zoo
Change into the source directory 'gap3r3p0/src/', print the list of
possible targets with the command 'make', select the target that fits
best, and then compile the kernel with the command
make <target>
cp gap ../bin
Alternatively we provide ready compiled executables for the some popular
systems on our 'ftp' server in the 'bin/' subdirectory.
Change into the directory 'gap3r3p0/bin/' and edit the script 'gap.sh',
which starts GAP, according to the instructions in this file. Then copy
this script to a directory in your search path, e.g., '~/bin/', with the
command
cp gap.sh ~/gap
When you later move GAP to another location you must only edit this
script.
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.
Change into the documentation directory 'gap3r3p0/doc/' and make the
printed manual with the commands (or similar commands, according to your
local custom for using LaTeX)
latex manual; latex manual; lp -dvi manual.dvi
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
suggestions.
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
forum.
To subscribe send a message to '[email protected]'
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
'[email protected]'.
'[email protected]' 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 technical or installation problems, we suggest that you write
to '[email protected]' instead of the GAP forum, as
such discussions are usually not very interesting for a larger audience.
Your e-mail message will be read by several people here, and we shall try
to provide support.
Thank you for your attention, Martin.
-- .- .-. - .. -. .-.. --- ...- . ... .- -. -. .. -.- .-
Martin Sch"onert, [email protected], +49 241 804551
Lehrstuhl D f"ur Mathematik, Templergraben 64, RWTH, 52056 Aachen, Germany
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