On Groups with Finite Hirsch Number

Abstract

Suppose G is a group with finite Hirsch number h modulo the k-th term of its upper central series. The Hirsch number of the (k + 1)-th term of the lower central series of G is known to be finite and of order bounded in terms of h and k. Here we give simpler proofs leading to simpler and sharper bounds. In particular and perhaps surprisingly, we show that the Hirsch number of the (h + 2k + 1)-th term of the lower central series of G is bounded by h(h + 3)/2; in particular it is bounded independently of k

Keywords

• finite hirsch number
• finite torsion-free rank
• generalized soluble group
• finite group