Solution to the presentation problem for powers of the augmentation ideal of torsion free and torsion abelian groups

This paper solves for torsion free and torsion abelian groups G the problem of presenting nth powers Δⁿ(G) of the augmentation ideal Δ(G) of an integral group ring , in terms of the standard additive generators of Δⁿ(G). A concrete basis for Δⁿ(G) is obtained when G itself has a basis and is torsion. The results are applied to describe the homology of the sequence Δn(N)G?Δn(G)?Δn(G/N).