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Algorithms for matrix groups

Published online by Cambridge University Press:  05 July 2011

By
E. A. O'Brien
Edited by
C. M. Campbell ,
M. R. Quick ,
E. F. Robertson ,
C. M. Roney-Dougal ,
G. C. Smith  and
G. Traustason
E. A. O'Brien
Affiliation:
University of Auckland
C. M. Campbell
Affiliation:
University of St Andrews, Scotland
M. R. Quick
Affiliation:
University of St Andrews, Scotland
E. F. Robertson
Affiliation:
University of St Andrews, Scotland
C. M. Roney-Dougal
Affiliation:
University of St Andrews, Scotland
G. C. Smith
Affiliation:
University of Bath
G. Traustason
Affiliation:
University of Bath
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Summary

Abstract

Existing algorithms have only limited ability to answer structural questions about subgroups G of GL(d, F), where F is a finite field. We discuss new and promising algorithmic approaches, both theoretical and practical, which as a first step construct a chief series for G.

Introduction

Research in Computational Group Theory has concentrated on four primary areas: permutation groups, finitely-presented groups, soluble groups, and matrix groups. It is now possible to study the structure of permutation groups having degrees up to about ten million; Seress [97] describes in detail the relevant algorithms. We can compute useful descriptions for quotients of finitely-presented groups; as one example, O'Brien & Vaughan-Lee [90] computed a power-conjugate presentation for the largest finite 2-generator group of exponent 7, showing that it has order 720416. Practical algorithms for the study of polycyclic groups are described in [59, Chapter 8].

We contrast the success in these areas with the paucity of algorithms to investigate the structure of matrix groups. Let G = 〈X〉 ≤ GL(d, F) where F = GF(q). Natural questions of interest to group-theorists include: What is the order of G? What are its composition factors? How many conjugacy classes of elements does it have? Such questions about a subgroup of Sn, the symmetric group of degree n, are answered both theoretically and practically using highly effective polynomialtime algorithms.

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Groups St Andrews 2009 in Bath , pp. 297 - 323
Publisher: Cambridge University Press
Print publication year: 2011

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