Some properties of direct products

Abstract

always implies C NN D. The classical theorem of Remak 1161 states that every finite group G, G # 1 can be expressed as a direct product of non trivial directly indecomposable groups in essentially one way. That is, in any two such decompositions, there is a one to one correspondence between the terms in each decomposition with corresponding terms isomorphic. Consequently cancellation in finite groups G is a trivial matter. A natural question is that of determining conditions to guarantee the cancellability of a group M. Over the last few decades various authors have considered this problem. A brief browsing through [4 J, for example, reveals concern for these questions in abelian groups. Related studies have been made on refinements of direct decompositions and attempts to obtain certain invariants. A partial listing (by no means complete) of these works is included in the reference of [lo]. These results are too numerous to elaborate on in this introduction. However, perhaps one can get some inkling of the inherent difficulties by considering the following sample results: