The associated graded ring of an integral group ring

Passi, Inder Bir S. ; Vermani, Lekh Raj (1977) The associated graded ring of an integral group ring Mathematical Proceedings of the Cambridge Philosophical Society, 82 (1). pp. 25-33. ISSN 0305-0041

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Let G be an Abelian group, S(G)=∑n≥0SPn(G) the symmetric algebra of G and grZG=∑n≥0Qn(G) the associated graded ring of the integral group ring ZG, where Qn(G)=AGN/AGn+1(AG (AG 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 θn(x1⊗^....⊗^xn=(xn-1)....(xn-1)+AGn+1 (xnεG). 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 θn for finite Abelian groups. In view of proposition 3.25 of (3) it is enough to consider finite Abelian p-groups.

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