A basis for augmentation quotients of finite abelian groups

Let G be a finite abelian group, ZG its associated integral group ring, and Δ(G) its augmentation ideal. In this paper we determine an explicit basis for the consecutive quotient groups Δn(G)/Δn+1(G) for any positive integer n and thereby compute precisely each of these quotient groups. This settles completely a problem of Karpilovsky.