Rank Restrictions, Varietal Properties and Wreath Products

Abstract

This chapter is a bit of a hotch-potch. In the first part we give an account of Platonov’s proof [43] of the nilpotence of the Frattini subgroup of a finitely generated linear group (4.17). Its main ingredient is a generalization (10.4) of Mal’cev’s Theorem (4.2) on the residual finiteness of a finitely generated linear group. This result is, I think, of independent interest and may well have many applications yet to be discovered. It will enable us to give, for example, an elementary proof of another result of Platonov on linear groups of finite rank ([46 b]). We also include in this section some simple structure theorems, taken from [69 c], for linear groups satisfying certain 2-generator solubility conditions.