Some infinite soluble groups, their modules, and the arithmeticity of associated automorphism groups.
type: Item
instance of: scholarly article
title: Some infinite soluble groups, their modules, and the arithmeticity of associated automorphism groups.
zbMATH Open document ID: 1070.20002
author: Christopher J. B. Brookes
author: J. R. J. Groves
published in: Journal of Algebra
publication date: 2005-02-22T00:00:00Z
review text: In recent years the authors have made detail studies of modules over a crossed product of a division ring by an Abelian group, promising us that there were group theoretic applications to come. In this current paper we get these applications, or at least some of them. There are three. The first concerns a module \(M\) over the group ring \(RG\), where \(R\) is a commutative ring and \(G\) is a torsion-free `connected' polycyclic group and gives conditions under which \(M\) is torsion-free over a sizeable section of \(RG\). Connected here means that the group can be embedded as a Zariski-connected subgroup of some \(\text{GL}(n,\mathbb Z)\). The second, with \(M\), \(R\) and \(G\) as above, but now with \(R\) Noetherian, \(M\) finitely generated and \(G\) nilpotent, gives conditions under which a specific subgroup of \(\Aut G\) the authors denote by \(\text{Stab}_{\Aut G}M\) is arithmetic. The third concerns a finitely generated nilpotent-nilpotent-by-finite group \(G\) with Fitting subgroup \(F\) such that no subgroup of \(G\) of finite index has a quotient isomorphic to a wreath product of a cyclic group of prime order by an infinite cyclic group. They show that the group of automorphisms induced on \(G/F\) by \(\Aut G\) has a subgroup of finite index acting nilpotently on \(G/F\) and that any finitely generated extension of \(G\) by a nilpotent group is again nilpotent-by-nilpotent-by-finite.
Mathematics Subject Classification ID: 20C07
Mathematics Subject Classification ID: 20F16
Mathematics Subject Classification ID: 20F28
Mathematics Subject Classification ID: 20F18
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 16S34
zbMATH DE Number: 2136593
zbMATH Keywords: modules
zbMATH Keywords: infinite soluble groups
zbMATH Keywords: automorphism groups
zbMATH Keywords: group rings
zbMATH Keywords: polycyclic groups
zbMATH Keywords: nilpotent groups
reviewed by: B. A. F. Wehrfritz
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2004.01.018
OpenAlex ID: W2006495750
cites work: Q3809999
cites work: Primitive Group Rings and Noetherian Rings of Quotients
cites work: Injective Modules, Induction Maps and Endomorphism Rings
cites work: Modules Over Nilpotent Group Rings
cites work: Modules over crossed products of a division ring by a free Abelian group. I
cites work: Modules over crossed products of a division ring by a free Abelian group. II
cites work: Q3994888
cites work: HNN-extensions of finitely presented soluble groups
cites work: Observations on group rings
cites work: Group rings of polycyclic groups
cites work: Prime Ideals in Group Rings of Polycyclic Groups
cites work: On the Residual Simplicity of Certain Modules
cites work: Q3663506
cites work: Completely reducible infinite-dimensional skew linear groups
DOI: 10.1016/J.JALGEBRA.2004.01.018
Recommended article: On soluble groups of module automorphisms of finite rank
Recommended article: Alcuni recenti contributi alla teoria dei gruppi infiniti risolubili
Recommended article: On finite soluble groups and the fixed-point groups of automorphisms
Recommended article: Q3593240
Recommended article: Q4386815
Recommended article: Q5260227
Recommended article: Q4664538
Recommended article: On finite soluble groups with almost fixed-point-free automorphisms of noncoprime order.
description: scientific article; zbMATH DE number 2136593