Related papers: Left 3-Engel elements in groups

In this paper we study left and right 4-Engel elements of a group. In particular, we prove that $<a, a^b>$ is nilpotent of class at most 4, whenever $a$ is any element and $b^{\pm 1}$ are right 4-Engel elements or $a^{\pm 1}$ are left…

Let $G$ be a group and let $x\in G$ be a left $3$-Engel element of order dividing $60$. Suppose furthermore that $\langle x\rangle^{G}$ has no elements of order $8$, $9$ and $25$. We show that $x$ is then contained in the locally nilpotent…

For any odd prime $p$, we give an example of a locally finite $p$-group $G$ containing a left 3-Engel element $x$ where $\langle x \rangle^G$ is not nilpotent.

In this paper we prove a group theoretic analogue of the well known local nilpotence theorem for sandwich Lie algebras due to Kostrikin and Zel'manov. We introduce the notion of a strong left 3-Engel element of a group G and show that these…

We give an example of a locally nilpotent group $G$ containing a left $3$-Engel element $x$ where $\langle x\rangle^{G}$ is not nilpotent.

We prove that the set of right 4-Engel elements of a group $G$ is a subgroup for locally nilpotent groups $G$ without elements of orders 2, 3 or 5; and in this case the normal closure $<x>^G$ is nilpotent of class at most 7 for each right…

We give an infinite family of examples that generalise the construction given in arXiv:1811.12074 of a locally finite 2-group $G$ containing a left 3-Engel element $x$ where ${\langle x \rangle}^G$, the normal closure of $x$ in $G$, is not…

Let $R_n(G)$ denotes the set of all right $n$-Engel elements of a group $G$. We show that in any group $G$ whose 5th term of lower central series has no element of order 2, $R_3(G)$ is a subgroup. Furthermore we prove that $R_4(G)$ is a…

We prove that 5-Engel Lie algebras over a field of characteristic zero, or over a field of prime characteristic $p>7$, are nilpotent of class at most 11. We also prove that if $G$ is a finite 5-Engel $p$-group for $p>7$ then $G$ is…

In this paper we study powerful 3-Engel groups. In particular, we find sharp upper bounds for the nilpotency class of powerful 3-Engel groups and the subclass of powerful metabelian 3-Engel groups.

We survey left 3-Engel elements in groups.

In my article 5-Engel algebras published on the arXiv in 2023 I proved that 5-Engel Lie algebras of characteristic zero or prime characteristic $p>7$ are nilpotent of class at most 11. In this note I investigate the ideal ID$(x)$ generated…

A classical theorem of R. Baer describes the nilpotent radical of a finite group G as the set of all Engel elements, i.e. elements y in G such that for any x in G the n-th commutator [x,y,...,y] equals 1 for n big enough. We obtain a…

We give an affrmative 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…

We study properties of Engel elements in weakly branch groups, lying in the group of automorphisms of a spherically homogeneous rooted tree. More precisely, we prove that the set of bounded left Engel elements is always trivial in weakly…

Let $q$ be a prime. Let $G$ be a residually finite group satisfying an identity. Suppose that for every $x \in G$ there exists a $q$-power $m=m(x)$ such that the element $x^m$ is a bounded Engel element. We prove that $G$ is locally…

We give a survey of results on the structure of right and left Engel elements of a group. We also present some new results in this topic.

Let m,n be positive integers, v a multilinear commutator word and w=v^m. We prove that if G is an orderable group in which all w-values are n-Engel, then the verbal subgroup v(G) is locally nilpotent. We also show that in the particular…

Let $G$ be a profinite group in which for every element $x\in G$ there exists a natural number $q=q(x)$ such that $x^q$ is Engel. We show that $G$ is locally virtually nilpotent. Further, let $p$ be a prime and $G$ a finitely generated…

For an element $g$ of a group $G$, an Engel sink is a subset ${\mathscr E}(g)$ such that for every $x\in G$ all sufficiently long commutators $[...[[x,g],g],\dots ,g]$ belong to ${\mathscr E}(g)$. A~finite group is nilpotent if and only if…