Lie properties of symmetric elements in group rings. II.
type: Item
instance of: scholarly article
title: Lie properties of symmetric elements in group rings. II.
zbMATH Open document ID: 1170.16019
author: Gregory T. Lee
author: Sudarshan K. Sehgal
author: Ernesto Spinelli
published in: Journal of Pure and Applied Algebra
publication date: 2009-03-30T00:00:00Z
review text: Let \(F\) be a field of characteristic different from \(2\), and \(G\) a group with involution *. Let \((FG)^+\) denote the set of symmetric elements of \(FG\), i.e. the elements \(\alpha\in FG\) so that \(\alpha=\alpha^*\). In part I [J. Algebra 321, No. 3, 890-902 (2009; Zbl 1169.16014)] \textit{A. Giambruno, C. Polcino Milies}, and \textit{S. K. Sehgal} showed that if \(G\) has no \(2\)-elements, and \((FG)^+\) is Lie nilpotent (resp. Lie \(n\)-Engel), then \(FG\) is Lie nilpotent (resp. Lie \(m\)-Engel, for some \(m\)). In this paper the authors complete this result by classifying the groups containing \(2\)-elements such that \((FG)^+\) is Lie nilpotent or Lie \(n\)-Engel. The main results are as follows. Let \(F\) be a field of characteristic \(p>2\), and let \(G\) be a group with involution *. Theorem 1. Suppose that \(FG\) is not Lie nilpotent. Then \((FG)^+\) is Lie nilpotent if and only if \(G\) is nilpotent, and \(G\) has a finite normal *-invariant \(p\)-subgroup \(N\) such that \(G/N\) is an SLC-group. By definition, a group \(H\) endowed with an involution * is an SLC-group if \(H/Z(H)\) is the Klein four group (and thus \(G\) has a unique nonidentity commutator, say \(z\)) and \(g^*=g\) if \(g\in Z(G)\), and otherwise \(g^*=zg\). Theorem 2. Suppose that \(FG\) is not Lie \(m\)-Engel, for any \(m\). Then \((FG)^+\) is Lie \(n\)-Engel if and only if \(G\) is nilpotent, \(G\) has a \(p\)-Abelian *-invariant normal subgroup \(A\) of finite index, and \(G\) has a normal *-invariant \(p\)-subgroup \(N\) of bounded exponent, such that \(G/N\) is an SLC-group.
reviewed by: Eric Jespers
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 16W10
Mathematics Subject Classification ID: 20C07
Mathematics Subject Classification ID: 16R50
zbMATH DE Number: 5534986
zbMATH Keywords: group rings
zbMATH Keywords: Lie properties
zbMATH Keywords: group algebras
zbMATH Keywords: Lie nilpotent algebras
zbMATH Keywords: Lie \(n\)-Engel algebras
zbMATH Keywords: symmetric elements
zbMATH Keywords: involutions
zbMATH Keywords: skew-symmetric elements
MaRDI profile type: Publication
Wikidata QID: Q115345584
full work available at URL: j.jpaa.2008.11.027
OpenAlex ID: W2066901180
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DOI: 10.1016/J.JPAA.2008.11.027
description: scientific article; zbMATH DE number 5534986