Narrow Results

We give an affirmative answer to the question whether a residually finite Engel group satisfying an identity is locally nilpotent. More generally, for a residually finite group $G$ with an identity, we prove that the set of right Engel elements of $G$ is contained in the Hirsch–Plotkin radical of $G$. Given an arbitrary word $w$, we also show that the class of all groups $G$ in which the $w$-values are right $n$-Engel and $w(G)$ is locally nilpotent is a variety.

Let $G$ be a group, $F$ a field of characteristic $p\ge 0$, and ${𝒰}(FG)$ the unit group of the group algebra $FG$. In this paper, among other results, we show that if either (1) $FG$ satisfies a non-matrix polynomial identity, or (2) $G$ is locally finite, $F$ is infinite and ${𝒰}(FG)$ is an Engel-by-finite group, then the $p$-elements of $G$ form a (normal) subgroup $P$ and $G/P$ is abelian (here, of course, $P=1$ if $p=0$).

In this paper, we study some algebras $A$ whose unit groups ${𝒰}(A)$ or subnormal subgroups of ${𝒰}(A)$ are (generalized) Engel. For example, we show that any generalized Engel subnormal subgroup of the multiplicative group of division rings with uncountable centers is central. Some of algebraic structures of Engel subnormal subgroups of the unit groups of skew group algebras over locally finite or torsion groups are also investigated.

In this paper, by considering the notions of polygroup and Engel group, we introduce the concept of Engel fuzzy subpolygroups. In this regard, by a normal Engel fuzzy subpolygroup $\mu$ of $P$ and ${\beta }^{*}$, the fundamental relation on a given polygroup $P$, we construct an Engel fuzzy subgroup ${\mu }_{{\beta }^{*}}$. We obtain a necessary and sufficient condition between Engel fuzzy subpolygroups and the Engel group $P$/$\sim$, the group of equivalence classes derived from a fuzzy subpolygroup of $P$. Finally, by using these results, we get Zorn’s lemma, in the Engel fuzzy subpolygroups.