Bounded Engel and solvable unitary units in group rings
type: Item
instance of: scholarly article
title: Bounded Engel and solvable unitary units in group rings
zbMATH Open document ID: 1447.16037
author: Gregory T. Lee
author: Sudarshan K. Sehgal
author: Ernesto Spinelli
published in: Journal of Algebra
publication date: 2018-03-01T00:00:00Z
review text: Let \(FG\) be the group algebra of a group \(G\) over a field \(F\) and let \(U(FG)\) be the unit group of the group algebra \(FG\). Consider the classical involution \({\ast}\) on FG, given by \[ {\Big( \sum_{g\in G} \alpha_gg\Big)}^{\ast} = \sum_{g\in G} \alpha_gg^{-1},\\\ \alpha_g \in F. \] The set \(U_n(FG)=\{x\in U(FG): xx^{\ast}=1\}\) is a subgroup of \(U(FG)\) and it is called the group of unitary units of \(FG\). The authors recall that the group algebra \(FG\) is called modular if char\(F=p>0\) and \(G\) has an element of prime order \(p\) and that \(G\) is \(p\)-abelian if \(G\) is a finite \(p\)-group. It is well known when the group \(U(FG)\) is nilpotent [\textit{I. I. Hripta}, Math. Notes 11, 119--124 (1972; Zbl 0258.20005); translation from Mat. Zametki 11, 191--200 (1972); \textit{J. L. Fisher} et al., Proc. Am. Math. Soc. 59, 195--200 (1976; Zbl 0341.16001)] and [\textit{I. I. Khripta}, On the multiplicative group of a group ring. Uzhgorod (Diss.) (1971)]. The main results of the paper are connected with the above indicated results. They are the following. Theorem 1. Let \(F\) be an infinite field of characteristic \(p >\) 2 and \(G\) a group, such that \(FG\) is modular. Then the following are equivalent: 1. \(U_n(FG)\) is bounded Engel and solvable, 2. \(U(FG)\) is nilpotent, and 3. \(G\) is nilpotent and \(p\)-abelian. Theorem 2. Let \(F\) be an algebraically closed field of characteristic different from 2 and \(G\) a group, such that \(FG\) is not modular. Then the following are equivalent: 1. \(U_n(FG)\) is bounded Engel and solvable, 2. \(U(FG)\) is nilpotent, and 3. \(G\) is nilpotent and the torsion elements of \(G\) are central.
Mathematics Subject Classification ID: 16U60
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 20C07
Mathematics Subject Classification ID: 20D10
zbMATH DE Number: 6845989
zbMATH Keywords: group ring
zbMATH Keywords: unitary units
zbMATH Keywords: bounded Engel
zbMATH Keywords: solvable
reviewed by: Todor Mollov
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2017.12.021
OpenAlex ID: W2782099515
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DOI: 10.1016/J.JALGEBRA.2017.12.021
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description: scientific article; zbMATH DE number 6845989