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

TL;DR

This paper proves that profinite groups with a prime order automorphism, where fixed points have finite Engel sinks, contain large structured subgroups, advancing understanding of their internal composition.

Contribution

It establishes the existence of open subgroups with specific nilpotent properties in profinite groups under automorphisms with finite Engel sinks at fixed points.

Findings

Groups have open locally nilpotent subgroups with finite right Engel sinks.

Groups have open pronilpotent-by-nilpotent subgroups with finite left Engel sinks.

Automorphism conditions impose strong structural restrictions on profinite groups.

Abstract

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.