A basis for augmentation quotients of finite abelian groups

Abstract

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.

Stability in graded rings associated with commutative augmented rings

“…for any positive integer n. This problem has been well studied in [4]- [15] for integral group rings, in [2], [16]- [18] for representation rings, and in [19]- [24] for Burnside rings.…”

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].…”