# Profinite groups and the fixed points of coprime automorphisms

## TL;DR

This paper proves that a profinite group acted upon by a coprime elementary abelian group with locally nilpotent fixed points must itself be locally nilpotent, extending understanding of automorphism actions on profinite groups.

## Contribution

It establishes a new criterion for local nilpotency of profinite groups based on coprime automorphisms with locally nilpotent fixed points.

## Key findings

- Profinite groups with coprime automorphisms and locally nilpotent fixed points are locally nilpotent.
- The result generalizes previous theorems on automorphisms of finite groups to profinite groups.
- Provides a new tool for analyzing the structure of profinite groups under automorphism actions.

## Abstract

The main result of the paper is the following theorem. Let $q$ be a prime and $A$ an elementary abelian group of order $q^3$. Suppose that $A$ acts coprimely on a profinite group $G$ and assume that $C_G(a)$ is locally nilpotent for each $a\in A^{\#}$. Then the group $G$ is locally nilpotent.