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