Group rings with metabelian unit groups
type: Item
instance of: scholarly article
title: Group rings with metabelian unit groups
zbMATH Open document ID: 1500.16030
author: Tibor Juhász
author: Ernesto Spinelli
published in: Journal of Pure and Applied Algebra
publication date: 2022-01-21T00:00:00Z
review text: Let \(G\) be a group, \(F\) a field of positive characteristic \(p\) and denote by \(FG\) the group algebra of \(G\) over \(F\) and by \(U(FG)\) the unit group of \(FG\). Describing the structure of \(U(FG)\) is a notoriously difficult problem. Though in a series of papers the cases when \(U(FG)\) is solvable have been characterized, the last contribution being \textit{A. Bovdi}'s [Commun. Algebra 33, No. 10, 3725--3738 (2005; Zbl 1082.16036)], the derived length of \(U(FG)\) in these cases is not that well understood. For \(p\) odd and \(G\) finite \textit{A. Shalev} described when \(U(FG)\) is metabelian [J. Pure Appl. Algebra 72, No. 3, 295--302 (1991; Zbl 0735.16020)] and this has been generalized by \textit{F. Catino} and the second author to the case that \(G\) is a torsion group [J. Group Theory 13, No. 4, 577--588 (2010; Zbl 1205.16030)]. The result is that \(U(FG)\) is metabelian if and only if \(p=3\) and the derived subgroup \(G'\) is central of order \(p\). In this paper the authors generalize this description to all infinite groups, obtaining a somehow surprising result. Denote by \(\mathrm{Syl}_p(G)\) the maximal \(p\)-subgroup of \(G\), which is unique when \(G\) is nilpotent. Again assume that \(p\) is odd. Then \(U(FG)\) is metabelian if and only if \(G\) is nilpotent of class \(2\) and either \(p = 3\) and \(G'\) has order \(p\) or \(p = 3\) and \(G' = \mathrm{Syl}_p(G)\) is elementary abelian of order \(p^2\) or \(p = 5\) and \(G' = \mathrm{Syl}_p(G)\) has order \(p\). This is proven by employing general bounds on the derived length of \(U(FG)\) and then constructing elements in \(U(FG)\) which show that the derived length is bigger than \(2\) in the cases not included in the theorem switching calculations between group commutators in \(U(FG)\) and Lie commutators in \(FG\).
reviewed by: Leo Margolis
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 16U60
zbMATH DE Number: 7461214
zbMATH Keywords: group rings
zbMATH Keywords: unit group
zbMATH Keywords: derived length
MaRDI profile type: Publication
full work available at URL: j.jpaa.2021.106946
OpenAlex ID: W3211190476
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DOI: 10.1016/J.JPAA.2021.106946
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description: scientific article; zbMATH DE number 7461214