Some Topics in the Theory of Infinite Soluble Groups

Abstract

Throughout this section Z will denote the ring of rational integers. By Theorem 3.27 a soluble subgroup of the unimodular group GL (n, Z) is polycyclic, a result first found by Mal′cev. This fact led P. Hall to ask whether an arbitrary polycyclic group has a faithful representation as a subgroup of GL (n, Z) for a suitable n. Partial solutions to this problem were obtained by Čarin ([3], Theorem 6), P. Hall ([6], p. 58), Jennings ([2], Theorem 8.1), Learner ([1], Theorem 3) and Wang [1]. Finally in 1967, a positive solution was obtained by L. Auslander [1], whose proof involved methods from the theory of Lie groups. We present here a purely algebraic proof of Hall’s conjecture due to Swan [1].