Almost Engel finite and profinite groups

TL;DR

This paper investigates the structure of groups where certain subgroups generated by iterated commutators are finite or bounded, establishing conditions under which the group has a finite normal subgroup with a locally nilpotent quotient.

Contribution

It proves that profinite groups with finite generated commutator subgroups have a finite normal subgroup with a locally nilpotent quotient, extending the Wilson–Zelmanov theorem.

Findings

Profinite groups with finite $E_n(g)$ subgroups have a finite normal subgroup with a locally nilpotent quotient.

In finite groups, bounded $E_n(g)$ subgroup sizes imply a bounded order of the nilpotent residual.

The proof leverages the Wilson–Zelmanov theorem on Engel profinite groups.

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 $m$.