Linear Methods and Soluble Groups

Abstract

Linear methods have been used extensively for quantitative investigations of soluble groups; for soluble linear groups, bounds in terms of the degree are known for such invariants as the order, derived length, etc. In 1956, P. Hall and G. Higman developed these ideas to obtain upper bounds for the p-length of a p-soluble group G in terms of various invariants of the Sylow p-subgroup of G. There emerged from this a body of techniques which have come to be known as Hall-Higman methods. The present chapter is an introduction to these methods. They are given in an elementary form in § 1, which is already sufficient to solve the restricted Burnside problem of exponent 6.