Throughout this chapter F is a field of characteristic p ≥ O and D is a locally finite-dimensional division F-algebra. (That is dimFF[X] is finite for every finite subset X of D.) We study the subgroups of GL(n,D). If X is any finitely generated subgroup of GL(n,D) then m = (F[X]:F) is finite and X is isomorphic to a subgroup of GL(m,F). Thus GL(n,D) is locally linear over F in the obvious sense. Clearly then the properties of skew linear groups over D are closely related to the corresponding properties of linear groups over F.

The results of this chapter both depend and extend the results of Chapter 2 via the exercise after 2.5.14, and 2.3.1. However, apart from a brief first section on general techniques, we concentrate on solubility, nilpotence and related concepts. The foundations of this theory were laid by Zalesski in 1967. In spite of much subsequent work (Wehrfritz) one important problem raised by Zalesski remains open, namely, is every locally nilpotent subgroup of GL(n,D) hypercentral. (We know from 1.4.4 that this is false for arbitrary division rings.)

As usual if p is a prime a p′-group is a periodic group with no elements of order p. A O′-group simply is a periodic group.