13 - A theorem on free products of special abelian groups

Summary

Introduction

In a previous paper [1], the quotient groups of the lower central series Ḡ_n = G_n/G_n+1 were studied. There the group G was assumed to be a free product of a finite number of finitely generated Abelian groups and G_n denoted the nth subgroup of the lower central series of G. Here we give an improved proof of a complicated lemma which first appeared in [1] (in particular, Lemma 4.4 of [1]). The proof given here, especially for property (iii) of the conclusion of that lemma, is a significant simplification of that which appears in [1]. We observe that one of the consequences of Lemma 4.4 of [1] is to give a set of free generators for the lower central quotients in the case where the free factors are torsion free (i.e., G = J in the terminology of [1]. Moreover the free generators are the J-basic commutators also using the terminology of Definition 4.1 of [1]). The authors only thought of this simplification after the publication of [1]. Furthermore, our improved proof uses results from [2] and [3].

In this paper, we will employ the notation, terminology, results, references, and equations of [1]. Furthermore, the numbering, a.b, of any definition, equation, etc., of [1] will correspond here to a.b-I. For example, Lemma 2.1-I will mean Lemma 2.1 of [1].

Preliminaries

In order to carry out our goal, we need to give some preliminary machinery.