On uniqueness of direct decompositions of groups into directly indecomposable factors

Abstract

In this paper, we give a necessary and sufficient condition for certain groups G to allow an essentially unique direct decomposition into non-trivial directly indecomposable groups. For any such G we show that if G has at least two essentially distinct direct decompositions into non-trivial directly indecomposable groups, then we can find finitely generated free abelian groups F and F 1 and a directly indecomposable group B for which 1. (a) B × F is a direct factor of G × F 1 and 2. (b) B × F has at least two essentially distinct direct decompositions into non-trivial directly indecomposable factors. The torsion free rank of F 1 may be expressed in terms of certain invariants of G.