# Profinite groups with an automorphism of prime order whose fixed points have finite Engel sinks

Khukhro, Evgeny and Shumyatsky, Pavel (2021) Profinite groups with an automorphism of prime order whose fixed points have finite Engel sinks. Monatshefte für Mathematik . ISSN 0026-9255

Full content URL: https://doi.org/10.1007/s00605-021-01561-5

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A right Engel sink of an element $g$ of a group $G$ is a set ${\mathscr R}(g)$ such that for every $x\in G$ all sufficiently long commutators $[...[[g,x],x],\dots ,x]$ belong to ${\mathscr R}(g)$. (Thus, $g$ is a right Engel element precisely when we can choose ${\mathscr R}(g)=\{ 1\}$.) We prove that if a profinite group $G$ admits a coprime automorphism $\varphi$ of prime order such that every fixed point of $\varphi$ has a finite right Engel sink, then $G$ has an open locally nilpotent subgroup.
A left 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 a left Engel element precisely when we can choose ${\mathscr E}(g)=\{ 1\}$.) We prove that if a profinite group $G$ admits a coprime automorphism $\varphi$ of prime order such that every fixed point of $\varphi$ has a finite left Engel sink, then $G$ has an open pronilpotent-by-nilpotent subgroup.