On finite groups with automorphisms whose fixed points are Engel

TL;DR

This paper proves that if a finite group admits a coprime elementary abelian automorphism group with fixed points that are all n-Engel, then the entire group is boundedly Engel, extending understanding of automorphism actions on finite groups.

Contribution

It establishes a bounded Engel property for finite groups under specific automorphism conditions involving fixed points and coprimality.

Findings

Group G is k-Engel for some bounded k depending on n and q.

Automorphism groups with fixed points as n-Engel impose strong structural constraints.

The result generalizes previous work on automorphisms and Engel conditions.

Abstract

The main result of the paper is the following theorem. Let $q$ be a prime, $n$ a positive integer and $A$ an elementary abelian group of order $q^2$. Suppose that $A$ acts coprimely on a finite group $G$ and assume that for each $a\in A^{\#}$ every element of $C_{G}(a)$ is $n$-Engel in $G$. Then the group $G$ is $k$-Engel for some $\{n,q\}$-bounded number $k$.