Some Examples of Linear Groups

Abstract

In this chapter we shall have a look at what sort of group has a faithful representation of finite degree over a field. Our plan of campaign is roughly as follows. Firstly we consider abelian groups, and then soluble groups —especially those satisfying either the minimal condition or the maximal condition on subgroups. Next we prove a local theorem of Mal’cev, which states that a group has a faithful representation of degree n if and only if each of its finitely generated subgroups has such a representation. Then we prove that a free group has a faithful representation of degree 2 over most fields, and discuss similar results for certain relatively free groups. We follow this with Nisnevič’s Theorem, that a free product of groups having faithful representations of degree n over fields of characteristic p≥0, has a faithful representation of degree n +1 over some field of characteristic p, plus some related results. Finally we consider the representability of wreath products of linear groups.