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GROUP ALGEBRAS WITH ENGEL UNIT GROUPS

Part of: Rings and algebras arising under various constructions Rings with polynomial identity Representation theory of groups Special aspects of infinite or finite groups

Published online by Cambridge University Press:  16 March 2016

M. RAMEZAN-NASSAB
M. RAMEZAN-NASSAB*
Affiliation:
Department of Mathematics, Kharazmi University, 50 Taleghani St., Tehran, Iran School of Mathematics, Institute for Research in Fundamental Sciences (IPM), PO Box 19395-5746, Tehran, Iran email [email protected]
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Abstract

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Let $F$ be a field of characteristic $p\geq 0$ and $G$ any group. In this article, the Engel property of the group of units of the group algebra $FG$ is investigated. We show that if $G$ is locally finite, then ${\mathcal{U}}(FG)$ is an Engel group if and only if $G$ is locally nilpotent and $G^{\prime }$ is a $p$-group. Suppose that the set of nilpotent elements of $FG$ is finite. It is also shown that if $G$ is torsion, then ${\mathcal{U}}(FG)$ is an Engel group if and only if $G^{\prime }$ is a finite $p$-group and $FG$ is Lie Engel, if and only if ${\mathcal{U}}(FG)$ is locally nilpotent. If $G$ is nontorsion but $FG$ is semiprime, we show that the Engel property of ${\mathcal{U}}(FG)$ implies that the set of torsion elements of $G$ forms an abelian normal subgroup of $G$.

Keywords

group algebraEngel groupLie Engel ring

MSC classification

Primary: 16R50: Other kinds of identities (generalized polynomial, rational, involution)
Secondary: 16S34: Group rings , Laurent polynomial rings 20C07: Group rings of infinite groups and their modules 20F45: Engel conditions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 101 , Issue 2 , October 2016 , pp. 244 - 252
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

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