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GROUP ALGEBRAS WITH ENGEL UNIT GROUPS
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 101 / Issue 2 / October 2016
- Published online by Cambridge University Press:
- 16 March 2016, pp. 244-252
- Print publication:
- October 2016
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- Article
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- You have access
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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 
$G$.
