The Generation of the Lower Central Series

Abstract

Let G be a finite group with commutator subgroup G′. In an earlier paper (4) it was shown that each element of G′ is a product of n commutators, if 4n ≥ |G′|. The object of this paper is to improve this result in two directions:Theorem 1a. If (n + 2)!n! > 2|G′| — 2, then each element of G′ is a product of n commutators.Theorem 1b. If G is a p-group, with |G′| = pa, and if n(n + 1) > a, then each element of G′ is a product of n commutators.