Denote by D m the dihedral group of order 2m. Let R(D m ) be its complex representation ring, and let Δ(D m ) be its augmentation ideal. In this paper, we determine the isomorphism class of the n-th augmentation quotient Δ n (D m )/Δ n+1 (D m ) for each positive integer n.

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

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