Symmetric units and group identities
type: Item
instance of: scholarly article
title: Symmetric units and group identities
zbMATH Open document ID: 0910.16015
author: Sudarshan K. Sehgal
author: Antonio Giambruno
author: Angela Valenti
published in: Manuscripta Mathematica
publication date: 1999-04-14T00:00:00Z
review text: In the paper under review the authors discuss when the set of symmetric units of a group ring satisfies a group identity. A unit of a group algebra is called a symmetric unit if it is stable under the involution coming from the natural Hopf algebra structure of the group ring. Let \(G\) be a torsion group and let \(F\) be an infinite field of characteristic different from 2. The authors classify the groups \(G\) so that the symmetric units of the group algebra \(FG\) satisfy a group identity. The answer depends on the characteristic of the field and in case of finite, positive, characteristic, if the quaternion group \(K_8\) is a subgroup of \(G\). In more detail. The symmetric units satisfy a group identity if and only if the following holds: In the case when \(F\) is a field of characteristic 0 \(G\) is abelian or a Hamiltonian 2-group. In the case when \(F\) is of characteristic \(p\geq 3\), and \(FG\) satisfies a polynomial identity, and either if \(K_8\) is not a subgroup of \(G\) and then the derived group of \(G\) is of bounded exponent \(p^k\) for some \(k\geq 0\), or if \(K_8\leq G\) the set of \(p\)-elements of \(G\) forms a normal \(p\)-subgroup \(P\) of \(G\) so that \(G/P\) is a Hamiltonian 2-group and \(G\) is of bounded exponent \(4p^s\) for some \(s\geq 0\). The case of fields of characteristic 2 is not discussed.
Mathematics Subject Classification ID: 16U60
Mathematics Subject Classification ID: 16W10
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 20C07
zbMATH DE Number: 1208703
zbMATH Keywords: symmetric units
zbMATH Keywords: group algebras
zbMATH Keywords: involutions
zbMATH Keywords: polynomial identities
reviewed by: Alexander Zimmermann
MaRDI profile type: Publication
full work available at URL: s002290050076
OpenAlex ID: W1996203779
DOI: 10.1007/S002290050076
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description: scientific article; zbMATH DE number 1208703