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Dear GAP Forum,

This is the announcement of

GPL, the Group Presentations Library .

The GPL files

1. contain 27 presentations of sporadic groups (a sporadic group is a

sporadic simple group or its automorphism group). These presentations

come from [PS97]: C.E.Praeger and L.H.Soicher, "Low Rank Representations

and Graphs for Sporadic Groups", Cambridge University Press, 1997. Other

presentations may be added in due course.

2. contain lists of words generating many subgroups with respect to these

presentations.

3. give a general setup for storing presentations and subgroup generators.

4. enable its users to build finitely presented groups via strings.

5. enable its users to build string presentations from finitely

presented groups.

The presentations include those for all sporadic simple groups with

order up to that of the Higman-Sims group, and the automorphism groups

of these groups. In addition, with respect to these eleven

presentations, words are given for every maximal subgroup up to the

action of the automorphism group of the group presented. To achieve

this, words for several subgroups were found by the author. For rank > 5

maximal subgroups, this adds to the information given in [PS97].

The GPL distribution contains of two files:

gpl.g - the library and functions in GAP format

gpl.ps.gz - documentation in postscript format

The files are available from GAP's incoming directory:

math.rwth-aachen.de/pub/incoming/

and from the GPL home page:

http://www.maths.qmw.ac.uk/~rcl/GPL/GPL.html

http://www.can.nl/~lindenb/GPL/GPL.html

where online documentation is available as well.

Please send bugs and comments to

Roderik Lindenbergh

R.C.Lindenbergh@qmw.ac.uk

EXAMPLES

1. Using the library of group presentations.

gap> gap> Read("gpl.g"); gap> hs := GroupPresentationGPL("HS",1); GroupPresentation( "HS", 1 ) gap> InitializeGroupPresentationGPL( hs ); gap> sg := AvailableSubgroupsGPL(hs); [ M22, U3(5).2, U3(5), L3(4).2_1, A8.2, 2^4.s6, 4^3:psl(3,2), M11, 4.2^4:s5, 2xa6.2^2, 5:4xa5 ] gap> ops := OperationCosetsFpGroup( hs.group, sg[1] );; gap> List(ops.generators, x -> LargestMovedPointPerm(x)); [ 100, 100, 100, 100, 100, 100 ] gap>

2. Building 2xA5 via a string representing a coxeter graph.

gap> gap> 2dira5 := FpGroupViaStrings( "a3b5c" ); ## The map is: "abc" Group( f.1, f.2, f.3 ) gap> 2dira5.relators; [ f.1*f.2*f.1*f.2*f.1*f.2, f.2*f.3*f.2*f.3*f.2*f.3*f.2*f.3*f.2*f.3, f.1^2, f.2^2, f.3^2, f.1*f.3*f.1*f.3 ] gap>

3. Creating a compact string presentation for a coxeter group.

gap> gap> F5 := FreeGroup( "x1", "x2", "x3", "x4", "x5" );; gap> E1 := F5 / [ F5.1^2, F5.2^2, F5.3^2, F5.4^2, F5.5^2, > ( F5.1 * F5.3 )^2, ( F5.2 * F5.4 )^2, ( F5.1 * F5.2 )^3, > ( F5.2 * F5.3 )^3, ( F5.3 * F5.4 )^3, ( F5.4 * F5.1 )^3, > ( F5.1 * F5.5 )^3, ( F5.2 * F5.5 )^2, ( F5.3 * F5.5 )^3, > ( F5.4 * F5.5 )^2, > ( F5.1 * F5.2 * F5.3 * F5.4 * F5.3 * F5.2 )^2 ];; gap> CoxeterStringPresentation( E1 ); [ "A3B3C3D3A3E3C",, "(ABCDCB)^2" ]

INTRODUCTION AND MAIN FEATURES.

The Group Presentations Library (GPL) presents a new way to store

and use presentations and words generating subgroups in a given presentation.

The presentations and the words are stored as strings in a human readible

way. I tried to develop and document the package in such a way that it

is easy to add new words and presentations.

** Presentations and short words for subgroups for sporadic groups. **

The Group Presentations Library contains already 27 presentations for

sporadic groups. Numerous subgroups can be created with the stored short

words in the generators of the given presentations. The first chapter

of the documentation describes the functions that are programmed to

use the information available in the GPL library file. Besides, it

will give an overview of the available presentations and it will tell

for which subgroups in the available presentations words are given

in the GPL library file. Several new short words that generate subgroups

of sporadic groups were found by the author. Short words found by

Martin Schoenert were added as well to the list of words and

presentations that can be found in [PS97]. This last source of

information formed the first basis for this library file.

** Storing presentations and words. **

The code that builds the presentations and the subgroups of a group

in a given presentation from the GPL library file is quite general.

It is therefore possible to add other presentations and words in future

releases easily. If one wishes to use a new presentation on a one-off

basis, one can use the functions described in chapter 2. If a

presentation or a set of presentations is needed more regularly, we

could consider to add these presentations in future releases. Please

contact me via email or other means. Of course it is possible to add

presentations and words to your personal copy of the library file by

just following the format conventions described in the first chapter

and in the library file itself.

** Finitely presented groups versus string presentations. **

The second chapter describes an alternative approach to building

finitely presented groups inside GAP. The standard approach in GAP

is to use a set of abstract generators subject to a set of relations.

When the relations, which are words in the abstract generators, are

complicated, it is sometimes more convenient to represent some of

the relations in the form of a so-called Coxeter graph. In our

approach, functions are introduced that make it possible to enter

generators and implicit relations via Coxeter graphs and explicit

relations using strings. Functions that convert a finitely presented

group in such a string presentation are added as well.

** Thanks. **

I hereby thank Anton Cox and Julian Gilbey for their comments.

Almost last but not least I would like to give special thanks

to my supervisor, Dr. L.H. Soicher who showed me many techniques

for working with sporadic groups and GAP, and who carefully read

this documentation and used procedures on an early stage.

Finally, I would like to thank the European Union, which enabled me to do

this work, done during a period of eight months at Queen Mary and Westfield

college, London, via an HCM grant in Computational Group Theory.

Roderik Lindenbergh

School of Mathematical Sciences

Queen Mary & Westfield College

Mile End Road

London E1 4NS

R.C.Lindenbergh@qmw.ac.uk

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