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Trends in infinite dimensional linear groups

Published online by Cambridge University Press:  05 July 2011

By
Martyn R. Dixon ,
Leonid A. Kurdachenko ,
Jose M. Muñoz-Escolano  and
Javier Otal
Edited by
C. M. Campbell ,
M. R. Quick ,
E. F. Robertson ,
C. M. Roney-Dougal ,
G. C. Smith  and
G. Traustason
Martyn R. Dixon
Affiliation:
University of Alabama, U.S.A
Leonid A. Kurdachenko
Affiliation:
National University of Dnepropetrovsk, Ukraine
Jose M. Muñoz-Escolano
Affiliation:
University of Zaragoza, Spain
Javier Otal
Affiliation:
University of Zaragoza, Spain
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

In this paper we present a short survey discussing some recent results in the theory of infinite dimensional linear groups.

Introduction

In this paper R will denote a ring, G will denote a group and A will be a right RG–module. When R is a field, we shall denote it by F and, of course, A is then also a vector space over F. The group GL(F,A), of all F-automorphisms of A, and its subgroups, are called linear groups. Linear groups have played a very important role in algebra and other branches of mathematics. If dim FA (the dimension of A over F) is finite, n say, then a subgroup G of GL(F, A) is called a finite dimensional linear group. It is well known that in this case GL(F,A) can be identified with the group of all invertible n × n matrices with entries in F. The subject of finite dimensional linear groups is among the most studied branches of mathematics, having been built using the interplay between algebraic, geometrical, combinatorial and other methods. This theory is rich in many interesting and important results.

However, the study of the subgroups of GL(F,A) in the case when A has infinite dimension over F has been much more limited and normally requires some additional restrictions. One natural type of restriction to use here is a finiteness condition. The most fruitful example of such restrictions to date has undoubtedly been that of finitary linear groups.

Type
Chapter
Information
Groups St Andrews 2009 in Bath , pp. 271 - 282
Publisher: Cambridge University Press
Print publication year: 2011

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