The associated graded ring of an integral group ring

Abstract

Let G be an Abelian group, S(G)=∑_n≥0SPⁿ(G) the symmetric algebra of G and grZG=∑_n≥0Q_n(G) the associated graded ring of the integral group ring ZG, where Q_n(G)=A_G^N/A_Gⁿ⁺¹(A_G (A_G denotes the augmentation ideal of ZG). Then there is a natural epimorphism (4) θ (G): S(G) → grZG which is given on the nth component by θ_n(x₁⊗^....⊗^x_n=(x_n-1)....(x_n-1)+A_Gⁿ⁺¹ (x_nεG). In general θ is not an isomorphism. In fact Bachmann and Grunenfelder(1) have shown that for finite Abelian G, θ is an isomorphism if and only if G is cyclic. Thus it is of interest to investigate ker θ_n for finite Abelian groups. In view of proposition 3.25 of (3) it is enough to consider finite Abelian p-groups.