# On groups with automorphisms whose fixed points are Engel

## TL;DR

This paper investigates the structure of finite and profinite groups with automorphisms whose fixed points are Engel elements, proving conditions under which such groups are locally nilpotent or bounded Engel.

## Contribution

It completes the classification of groups with automorphisms having Engel fixed points, establishing new bounds and conditions for local nilpotency and Engel properties.

## Key findings

- Proves that groups with certain automorphism actions are locally nilpotent.
- Establishes bounds on the Engel degree for groups with automorphisms of specific orders.
- Provides conditions under which fixed points being Engel implies the entire group is Engel or nilpotent.

## Abstract

We complete the study of finite and profinite groups admitting an action by an elementary abelian group under which the centralizers of automorphisms consist of Engel elements. In particular, we prove the following theorems. Let $q$ be a prime and $A$ an elementary abelian $q$-group of order at least $q^2$ acting coprimely on a profinite group $G$. Assume that all elements in $C_{G}(a)$ are Engel in $G$ for each $a\in A^{\#}$. Then $G$ is locally nilpotent (Theorem B2). Let $q$ be a prime, $n$ a positive integer and $A$ an elementary abelian group of order $q^3$ acting coprimely on a finite group $G$. Assume that for each $a\in A^{\#}$ every element of $C_{G}(a)$ is $n$-Engel in $C_{G}(a)$. Then the group $G$ is $k$-Engel for some $\{n,q\}$-bounded number $k$ (Theorem A3).