On units of twisted group algebras
type: Item
instance of: scholarly article
title: On units of twisted group algebras
zbMATH Open document ID: 1006.16038
author: Chia-Hsin Liu
published in: Journal of Algebra
publication date: 2002-10-06T00:00:00Z
review text: This paper continues and extends the work of numerous authors on Hartley's problem. Specifically, let \(R\) be an algebra over a field \(K\) and suppose that its group of units \(U(R)\) does not contain a nonabelian free subgroup. The goal is to deduce information on the ring-theoretic structure of \(R\). Of course, it is natural to assume that \(R\) is generated by \(U(R)\), and hence group algebras and twisted group algebras are of particular interest. The main results here are: Theorem 1. Let \(K^t[G]\) be a twisted group algebra of the countable locally finite group \(G\) over the nonabsolute field \(K\) of characteristic \(p>0\). Then \(U(K^t[G])\) contains no nonabelian free subgroup if and only if \(G=PQ\), where \(P\) is a normal Sylow \(p\)-subgroup, \(P\cap Q=1\), and \(K^t[Q]\) is Abelian. Theorem 2. Let \(K^t[G]\) be a twisted group algebra of the locally finite group \(G\) over the infinite field \(K\) of characteristic \(p>0\). The following are equivalent: (1) \(U(K^t[G])\) satisfies a group identity; (2) \(K^t[G]\) satisfies a nonmatrix identity; (3) \(G'\) is a \(p\)-group of bounded period, \(G\) has a normal \(p\)-Abelian subgroup of finite index, and \(K^t[G]\) is stably untwisted; (4) \(U(K^t[G])\) satisfies \((x,y)^{p^k}=1\) for some integer \(k\geq 1\).
reviewed by: D. S. Passman
Mathematics Subject Classification ID: 16U60
Mathematics Subject Classification ID: 16S35
Mathematics Subject Classification ID: 16R50
Mathematics Subject Classification ID: 20F50
Mathematics Subject Classification ID: 20C07
zbMATH DE Number: 1762099
zbMATH Keywords: twisted group algebras
zbMATH Keywords: unit groups
zbMATH Keywords: locally finite groups
zbMATH Keywords: group identities
zbMATH Keywords: subgroups of finite index
MaRDI profile type: Publication
full work available at URL: jabr.2001.9088
OpenAlex ID: W2042232264
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DOI: 10.1006/JABR.2001.9088
description: scientific article; zbMATH DE number 1762099