# Engel-like conditions in fixed points of automorphisms of profinite groups

## TL;DR

This paper investigates conditions under which fixed points of automorphisms in profinite groups imply the group is locally virtually nilpotent, extending Engel conditions to automorphism actions of elementary abelian groups.

## Contribution

It establishes new criteria linking Engel conditions on automorphism fixed points to the local virtual nilpotency of profinite groups, generalizing previous results.

## Key findings

- Proves that automorphisms of order q^2 with Engel conditions imply local virtual nilpotency.
- Shows that automorphisms of order q^3 with Engel conditions on centralizers imply local virtual nilpotency.
- Provides quantitative analogues for finite groups related to these conditions.

## Abstract

Let $q$ be a prime and $A$ an elementary abelian $q$-group acting as a coprime group of automorphisms on a profinite group $G$. We show that if $A$ is of order $q^2$ and some power of each element in $C_G(a)$ is Engel in $G$ for any $a\in A^{\#}$, then $G$ is locally virtually nilpotent. Assuming that $A$ is of order $q^3$ we prove that if some power of each element in $C_G(a)$ is Engel in $C_G(a)$ for any $a\in A^{\#}$, then $G$ is locally virtually nilpotent. Some analogues consequences of quantitative nature for finite groups are also obtained.