# Almost Engel finite and profinite groups

Khukhro, E. I. and Shumyatsky, P. (2016) Almost Engel finite and profinite groups. [MIMS Preprint]

## Abstract

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 $[...[[x,g],g],\dots ,g]$ 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 $n$, $|E_n(g)|\leq m$ for all $g\in G$, then the order of the nilpotent residual $\gamma _{\infty}(G)$ is bounded in terms of $m$.

Item Type: MIMS Preprint MSC 2010, the AMS's Mathematics Subject Classification > 20 Group theory and generalizations Professor Evgeny Khukhro 08 Mar 2016 08 Nov 2017 18:18 http://eprints.maths.manchester.ac.uk/id/eprint/2446