Structure of augmentation quotients of finite homocyclic abelian groups

Abstract

Let G be a finite abelian group and its Sylow p-subgroup a direct product of copies of a cyclic group of order p r , i.e., a finite homocyclic abelian group. Let Δ n (G) denote the n-th power of the augmentation ideal Δ(G) of the integral group ring ZG. The paper gives an explicit structure of the consecutive quotient group Qn(G) = Δ n (G)/Δ n+1 (G) for any natural number n and as a consequence settles a problem of Karpilovsky for this particular class of finite abelian groups.

Augmentation Quotients for Burnside Rings of some Finite $p$-Groups

“…Two related problems of recent interest have been to investigate the augmentation ideals and their consecutive quotients for integral group rings and representation rings of finite groups. These problems have been well studied in [6], [7], [8], [9], [10], [11], [12], [13] and [14].…”

Augmentation quotients for burnside rings of generalized dihedral groups

“…Two related problems of recent interest have been to investigate the augmentation ideals and their consecutive quotients for integral group rings and representation rings of finite groups. These problems have been well studied in [1]- [5], [7]- [10].…”