On factorizations of finite abelian group which admit replacement of a Z‐set by a subgroup

Abstract

A subset A of a finite additive abelian group G is a Z-set if for all a A, na A for all n Z. The purpose of this paper is to prove that for a special class of finite abelian groups, whenever the factorization G A B, where A and B are Z-sets, arises from the series G K m K 2 Kn then there exist subgroups S and T such that the factorization G S T also arises from this series. This result is obtained through the introduction of two new concepts: a series admits replacement and the extendability of a subgroup. A generalization of a result of L. Fuchs is given which enables establishment of a necessary and sufficient condition for extendability. This condition is used to show that certain series for finite abelian p-groups admit replacement.