Star-group identities and groups of units.
type: Item
instance of: scholarly article
title: Star-group identities and groups of units.
zbMATH Open document ID: 1216.16024
author: Antonio Giambruno
author: Sudarshan K. Sehgal
author: C. Polcino Milies
published in: Archiv der Mathematik
publication date: 2011-01-07T00:00:00Z
review text: This paper is an extension of the authors' [J. Algebra 322, No. 8, 2801-2815 (2009; Zbl 1193.16027)]. Analogously to a star identity in rings with involution a star identity in groups is defined as an element of the free group of countable rank with involution evaluating identically to 1 for elements of a group \(G\) with involution. Suppose that \(G\) is a torsion group with involution * and that \(F\) is an infinite field with \(\text{char\,}F\neq 2\). Extend * linearly to \(FG\). The main result is the following. (1) If \(FG\) is semiprime then \(U\) satisfies a star identity if and only if \(G\) is Abelian or \(G\) is an SLC-group, i.e., \(G\) modulo the center is the Klein Four group. (2) If \(FG\) is not semiprime then \(U\) satisfies a star identity if and only if \(P\), the set of \(p\)-elements of \(G\), is a subgroup, \(FG\) satisfies a polynomial identity and one of the following holds: (i) \(G/P\) is Abelian and \(G'\) is of bounded \(p\)-power exponent; (ii) \(G/P\) is SLC and \(G\) contains a normal *-invariant \(p\)-subgroup \(B\) of bounded exponent such that \(P/B\) is central in \(G/B\) and * is trivial on \(P/B\).
reviewed by: János Kurdics
Mathematics Subject Classification ID: 16U60
Mathematics Subject Classification ID: 20C07
Mathematics Subject Classification ID: 16W10
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 16R50
zbMATH DE Number: 5833787
zbMATH Keywords: group identities
zbMATH Keywords: involutions
zbMATH Keywords: group algebras
zbMATH Keywords: star identities
zbMATH Keywords: rings with involution
zbMATH Keywords: polynomial identities
zbMATH Keywords: groups of units
MaRDI profile type: Publication
full work available at URL: s00013-010-0195-0
OpenAlex ID: W2108344014
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DOI: 10.1007/S00013-010-0195-0
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description: scientific article; zbMATH DE number 5833787