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Computing with matrix groups over infinite fields

Published online by Cambridge University Press:  05 July 2011

By
A. S. Detinko ,
B. Eick  and
D. L. Flannery
Edited by
C. M. Campbell ,
M. R. Quick ,
E. F. Robertson ,
C. M. Roney-Dougal ,
G. C. Smith  and
G. Traustason
A. S. Detinko
Affiliation:
National University of Ireland, Ireland
B. Eick
Affiliation:
Institut Computational Mathematics, Germany
D. L. Flannery
Affiliation:
National University of Ireland, Ireland
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

We survey currently available algorithms for computing with matrix groups over infinite domains. We discuss open problems in the area, and avenues for further development.

Introduction

The subject of linear groups is one of the main branches of group theory. Linear groups provide a link between group theory and natural sciences such as physics, chemistry, and genetics; as well as other areas of mathematics, including geometry, combinatorics, functional analysis, and differential equations.

The significance of linear groups was realized at the very beginning of group theory, dating back to work by C. Jordan (1870). In the early twentieth century, major successes in linear group theory were achieved by Burnside, Schur, Blichfeldt, and Frobenius; their results continue to exert an influence up to the present day.

Linear groups arise in various ways in the theory of abstract groups. For instance, they occur as groups of automorphisms of certain abelian groups, and they play a central role in the study of solvable groups. Furthermore, linearity is a vital property for some classes of groups: polycyclic-by-finite groups and countable free groups are prominent examples. Linear groups are closely associated to Lie groups, algebraic groups, and representation theory. For extra background we refer to [16, 48, 49, 50].

Advances in computational algebra have motivated a new phase in linear group theory. Matrix representations of groups have the advantage that a large (even infinite) group can be defined by input of small size.

Type
Chapter
Information
Groups St Andrews 2009 in Bath , pp. 256 - 270
Publisher: Cambridge University Press
Print publication year: 2011

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References

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