Some remarks on the unit groups of integral group rings
type: Item
instance of: scholarly article
title: Some remarks on the unit groups of integral group rings
zbMATH Open document ID: 0576.16011
DOI: 10.1007/BF01235773
author: Ashwani K. Bhandari
published in: Archiv der Mathematik
publication date: 1985-01-01T00:00:00Z
review text: Let G be a group. We denote by \(U_ 1({\mathbb{Z}}G)\) the set of units in \({\mathbb{Z}}G\) which have augmentation one. This paper studies the Lie residual nilpotence of \(\Delta_{{\mathbb{Z}}}(U_ 1{\mathbb{Z}}G)\). It is first shown that a group G is residually (nilpotent with derived group torsion free) if and only if \(U_ 1{\mathbb{Z}}G\) is. Then it is shown that, for an arbitrary group G, we have that \(\cap_{n}\Delta_{{\mathbb{Z}}}^{(n)}(G)=0\) if and only if \(\cap_{n}\Delta_{{\mathbb{Z}}}^{(n)}(U_ 1{\mathbb{Z}}G)=0\). Finally, the author deals with the residual solvability of U\({\mathbb{Z}}G\), a question raised by \textit{I. Musson} and \textit{A. Weiss} [Arch. Math. 38, 514-530 (1982; Zbl 0476.16010)]. He shows that if G is a group satisfying certain conditions on its commutator subgroup G' then \(U_ 1{\mathbb{Z}}G\) is residually solvable.
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 16U60
Mathematics Subject Classification ID: 20C07
Mathematics Subject Classification ID: 20E26
zbMATH DE Number: 3920706
zbMATH Keywords: units
zbMATH Keywords: Lie residual nilpotence
zbMATH Keywords: residual solvability
zbMATH Keywords: commutator subgroup
reviewed by: C. Polcino Milies
MaRDI profile type: Publication
cites work: Free Subgroups in the Unit Groups of Integral Group Rings
cites work: The residual nilpotency of the augmentation ideal and the residual nilpotency of some classes of groups
cites work: Integral group rings with residually nilpotent unit groups
cites work: Group rings and their augmentation ideals
cites work: Solutions of an Equation in Abelian Groups
cites work: Q4198893
full work available at URL: https://doi.org/10.1007/bf01235773
OpenAlex ID: W1987183308
description: scientific article; zbMATH DE number 3920706