# Nilpotent residual and Fitting subgroup of fixed points in finite groups

## TL;DR

This paper investigates how fixed points of automorphisms in finite groups influence the group's structure, establishing bounds on certain subgroups based on properties of centralizers under automorphisms.

## Contribution

It provides new bounds on the nilpotent residual and Fitting subgroup of a finite group based on automorphism fixed points, extending understanding of automorphism actions.

## Key findings

- Bound on the order of the nilpotent residual of G in terms of fixed point properties.
- Bound on the index of the second Fitting subgroup based on centralizer properties.
- Results depend solely on a parameter m related to fixed point sizes.

## Abstract

Let $q$ be a prime and $A$ a finite $q$-group of exponent $q$ acting by automorphisms on a finite $q'$-group $G$. Assume that $A$ has order at least $q^3$. We show that if $\gamma_{\infty} (C_{G}(a))$ has order at most $m$ for any $a \in A^{\#}$, then the order of $\gamma_{\infty} (G)$ is bounded solely in terms of $m$. If the Fitting subgroup of $C_{G}(a)$ has index at most $m$ for any $a \in A^{\#}$, then the second Fitting subgroup of $G$ has index bounded solely in terms of $m$.