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

Khukhro, Evgeny and Shumyatsky, Pavel (2021) On profinite groups with automorphisms whose fixed points have countable Engel sinks. Israel Journal of Mathematics . ISSN 0021-2172

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## 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.

Keywords: profinite groups, pro-$p$ groups, Lie ring method, Engel condition, locally nilpotent groups, automorphisms G Mathematical and Computer Sciences > G110 Pure Mathematics College of Science > School of Mathematics and Physics 44312 16 Apr 2021 10:03

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