GROUP ALGEBRAS WITH ENGEL UNIT GROUPS

• Published in: Journal of the Australian Mathematical Society (2001) 2016-10, Vol.101 (2), p.244-252
• Main Author: RAMEZAN-NASSAB, M.
• Format: Article
• Language: English
• ISSN: 1446-7887; 1446-8107
• DOI: 10.1017/S1446788716000094

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 $G$ .