On profinite groups with automorphisms whose fixed points have countable Engel sinks

TL;DR

The paper proves that a profinite group with certain automorphisms, where elements have countable Engel sinks, has a finite normal subgroup with a locally nilpotent quotient, extending understanding of Engel properties in automorphism actions.

Contribution

It establishes a new connection between automorphism groups with countable Engel sinks and the structural decomposition of profinite groups.

Findings

Existence of a finite normal subgroup in the group.

The quotient group is locally nilpotent.

Automorphisms with countable Engel sinks influence group structure.

Abstract

An Engel sink of an element $g$ of a group $G$ is a set ${\mathscr E}(g)$ such that for every $x\in G$ all sufficiently long commutators $[...[[x,g],g],\dots ,g]$ belong to ${\mathscr E}(g)$. (Thus, $g$ is an Engel element precisely when we can choose ${\mathscr E}(g)=\{ 1\}$.) It is proved that if a profinite group $G$ admits an elementary abelian group of automorphisms $A$ of coprime order $q^2$ for a prime $q$ such that for each $a\in A\setminus\{1\}$ every element of the centralizer $C_G(a)$ has a countable (or finite) Engel sink, then $G$ has a finite normal subgroup $N$ such that $G/N$ is locally nilpotent.