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

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