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Abstract. Let D be a non-commutative division ring, m and n two natural numbers, M a maximal and N a subnormal subgroup of GL m .D/. In this paper, among other results, we show that: (1) if N is an n-Engel group, then it is central; (2) if m > 1 and M is locally nilpotent, then M is abelian; (3) if m > 1 and M is n-Engel, then the Hirsch-Plotkin radical of M is abelian. Also, we define some generalized Engel conditions on groups, and then we prove similar results, as quoted, for these groups.

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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 $G$.