Group Algebras With Engel Unit Groups

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 $G$.

Group Algebras Whose Unit Group Is Locally Nilpotent

“…In several articles, Ramezan-Nassab attempted to describe the structure of groups G for which V(FG) are Engel (locally nilpotent) groups in the case when FG have only a finite number of nilpotent elements (see [ The next two theorems completely describe groups G with V(FG) locally nilpotent. Some special cases of Theorem 1.2 were proved by Khripta (see [16]) and Ramezan-Nassab (see [19,Theorem 1.2] and [20, Corollary 1.3 and Theorem 1.4]). Theorem 1.2.…”

Group algebra whose unit group is locally nilpotent

“…The next two theorems completely describe groups G with V (F G) locally nilpotent. Some special cases of Theorem 3 were proved by I. Khripta (see [17]) and M. Ramezan-Nassab (see Theorem 1.2 in [20] and Corollary 1.3 and Theorem 1.4 in [21]). Theorem 3.…”

“…In several articles, M. Ramezan-Nassab attempted to describe the structure of groups G for which V (F G) are Engel (locally nilpotent) groups in the case when F G have only a finite number of nilpotent elements (see Theorem 1.5 in [21], Theorems 1.2 and 1.3 in [20] and Theorem 1.3 in [22]). The following theorem gives a complete answer.…”