> < ^ Date: Thu, 27 Feb 1997 09:07:36 +0000
^ From: Roderik Cornelis Lindenbergh <r.c.lindenbergh@qmw.ac.uk >
^ Subject: Group Presentations Library

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
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:
and from the GPL home page:
where online documentation is available as well.

Please send bugs and comments to

Roderik Lindenbergh



1. Using the library of group presentations.

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 ]

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

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 ]

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

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" ] 


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


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