Profinite groups with an automorphism whose fixed points are right Engel

TL;DR

This paper proves that a profinite group with a coprime automorphism of prime order, whose fixed points are all right Engel elements, must be locally nilpotent, revealing a structural property under these conditions.

Contribution

It establishes a new link between automorphism fixed points being right Engel and the local nilpotency of profinite groups.

Findings

Profinite groups with such automorphisms are locally nilpotent.

Fixed points being right Engel elements imply strong structural constraints.

The result extends understanding of automorphism actions on profinite groups.

Abstract

An element $g$ of a group $G$ is said to be right Engel if for every $x\in G$ there is a number $n=n(g,x)$ such that $[g,{}_{n}x]=1$. We prove that if a profinite group $G$ admits a coprime automorphism $\varphi$ of prime order such that every fixed point of $\varphi$ is a right Engel element, then $G$ is locally nilpotent.