Solution to the presentation problem for powers of the augmentation ideal of torsion free and torsion Abelian groups.
type: Item
instance of: scholarly article
title: Solution to the presentation problem for powers of the augmentation ideal of torsion free and torsion Abelian groups.
zbMATH Open document ID: 1068.16032
author: Anthony Bak
author: Guoping Tang
published in: Advances in Mathematics
publication date: 2004-11-26T00:00:00Z
review text: Let \(\mathbb{Z} G\) be the integral group ring of the group \(G\) and let \(\Delta(G)\) be the augmentation ideal of \(\mathbb{Z} G\), i.e. \(\Delta(G)\) is the free Abelian group on the elements \(g-1\) (\(g\in G\)). Then the \(n\)-th power \(\Delta^n(G)\) of \(\Delta(G)\) is generated as an Abelian group by the standard generators \((g_1-1)(g_2-1)\cdots(g_n-1)\) with \(g_1,g_2,\dots,g_n\in G\). It is a classical problem in the theory of group rings to find all relations among the standard generators of \(\Delta^n(G)\). The authors solve this problem when \(G\) is a torsion free or torsion Abelian group. The results are applied to describe the homology of the sequence \(\Delta^n(N)G\to\Delta^n(G)\to\Delta^n(G/N)\), where \(G\) is an Abelian group and \(N\subseteq G\) is a subgroup.
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 20C07
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20F14
zbMATH DE Number: 2119092
zbMATH Keywords: integral group rings
zbMATH Keywords: augmentation ideals
zbMATH Keywords: Abelian torsion groups
zbMATH Keywords: torsion-free Abelian groups
zbMATH Keywords: presentations
zbMATH Keywords: homology
reviewed by: S. V. Mihovski
MaRDI profile type: Publication
full work available at URL: j.aim.2003.11.002
OpenAlex ID: W2004333072
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DOI: 10.1016/J.AIM.2003.11.002
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description: scientific article; zbMATH DE number 2119092