Narrow Results

Let $g$ be an element of a group $G$. For a positive integer $n$, let $E_n(g)$ be the subgroup generated by all commutators $[\dots [[x,g],g],\dots ,g]$ over $x\in G$, where $g$ is repeated $n$ times. We prove that if $G$ is a profinite group such that for every $g\in G$ there is $n=n(g)$ such that $E_n(g)$ is finite, then $G$ has a finite normal subgroup $N$ such that $G/N$ is locally nilpotent. The proof uses the Wilson–Zelmanov theorem saying that Engel profinite groups are locally nilpotent. In the case of a finite group $G$, we prove that if, for some $n$, $|E_n(g)|\le m$ for all $g\in G$, then the order of the nilpotent residual ${\gamma }_{\infty }(G)$ is bounded in terms of $m$.

Let R be an associative ring and let x, y ∈ R. Define the generalized commutators as follows: [x, ₀y] = x and [x, _ky] = [x, _k-1y]y - y[x, _k-1y](k = 1, 2, …). In this paper we study some generalized Engel rings, i.e. -rings (satisfying [x^{m(x, y)}, _{k(x, y)}y] = 0), -rings (satisfying [x^{m(x, y)}, _{k(x, y)}y^{n(x, y)}] = 0) and -rings (satisfying [x^{m(x, y)}, _{k(x, y)}y^{n(x, y)}]^{r(x, y)} = 0). Among other results, it is proved that every Artinian -ring is strictly Lie-nilpotent. Also, we show that in each of the following cases R has nil commutator ideal: (1) if R is a -ring with unity and k, n independent of y; (2) if R is a locally bounded -ring (defined below); (3) if R is an algebraic algebra over a field in which R* is a bounded Engel group or a soluble group.

Let $G$ be a group and $h,g\in G$. The 2-tuple $(h,g)$ is said to be an $n$-Engel pair, $n\ge 2$, if $h=[h{,}_{n}g]$, $g=[g{,}_{n}h]$ and $h\ne 1$. Let SL$(2,F)$ be the special linear group of degree $2$ over the field $F$. In this paper, we show that given any field $L$, there is a field extension $F$ of $L$ with $[F:L]\le 6$ such that SL$(2,F)$ has an $n$-Engel pair for some integer $n\ge 4$. We will also show that SL$(2,F)$ has a $5$-Engel pair if $F$ is a field of characteristic $p\equiv \pm 1mod5$.

Let $R$ be a noncommutative division ring with center $Z$, which is algebraic, that is, $R$ is an algebraic algebra over the field $Z$. Let $f$ be an antiautomorphism of $R$ such that (i) ${[f(x),x^{m(x)}]}_{n(x)}=0$, all $x\in R$, where $m(x)$ and $n(x)$ are positive integers depending on $x$. If, further, $f$ has finite order, it was shown in [M. Chacron, Antiautomorphisms with quasi-generalised Engel condition, J. Algebra Appl. 17(8) (2018) 1850145 (19 pages)] that $f$ is commuting, that is, $[f(x),x]=0$, all $x\in R$. Posed in [M. Chacron, Antiautomorphisms with quasi-generalised Engel condition, J. Algebra Appl. 17(8) (2018) 1850145 (19 pages)] is the question which asks as to whether the finite order requirement on $f$ can be dropped. We provide here an affirmative answer to the question. The second major result of this paper is concerned with a nonnecessarily algebraic division ring $R$ with an antiautomorphism $f$ satisfying the stronger condition (ii) ${[f(x),x^m]}_n=0$, all $x\in R$, where $m$ and $n$ are fixed positive integers. It was shown in [T.-K. Lee, Anti-automorphisms satisfying an Engel condition, Comm. Algebra 45(9) (2017) 4030–4036] that if, further, $f$ has finite order then $f$ is commuting. We show here, that again the finite order assumption on $f$ can be lifted answering thus in the affirmative the open question (see Question 2.11 in [T.-K. Lee, Anti-automorphisms satisfying an Engel condition, Comm. Algebra 45(9) (2017) 4030–4036]).

An Engel sink of an element $g$ of a group $G$ is a set $\mathcal{ℰ}(g)$ such that for every $x\in G$ all sufficiently long commutators $[...[[x,g],g],\dots ,g]$ belong to $\mathcal{ℰ}(g)$. (Thus, $g$ is an Engel element precisely when we can choose $\mathcal{ℰ}(g)=\{1\}$.) It is proved that if every element of a compact (Hausdorff) group $G$ has a countable (or finite) Engel sink, then $G$ has a finite normal subgroup $N$ such that $G/N$ is locally nilpotent. This settles a question suggested by J. S. Wilson.

The aim of this article is to survey some results and problems concerning the adjoint group of radical rings and related topics. A main attention is focused on the structure of this group, on the relationship between the adjoint and Lie structures in such rings and on the structure of radical rings whose adjoint group is finitely generated. Furthemore, some generalizations of radical rings to so-called braces and radical modules and their natural connection with triply factorized groups are considered. The article also contains proofs of some crucial results.