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 this 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 Schönert 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.
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
lindenbe@math.ruu.nl