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GROUP ALGEBRAS WITH ENGEL UNIT GROUPS
Published online by Cambridge University Press: 16 March 2016
Abstract
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
$G$.
- 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.
References
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