Frobenius groups of automorphisms with almost fixed point free kernel

TL;DR

This paper investigates the structure of finite solvable groups under the action of Frobenius automorphism groups, establishing bounds on the group's Fitting series based on fixed point subgroup sizes, generalizing previous results.

Contribution

It extends prior work by providing bounds on the index of the Fitting subgroup in terms of fixed point subgroup sizes for higher Fitting lengths.

Findings

Bound on the index of Fitting subgroup in terms of fixed points.

Generalization of previous results for Fitting length n.

Applicable to coprime automorphism actions on solvable groups.

Abstract

Let $FH$ be a Frobenius group with kernel $F$ and complement $H$, acting coprimely on the finite solvable group $G$ by automorphisms. We prove that if $C_{G}(H)$ is of Fitting length $n$ then the index of the $n$-th Fitting subgroup $F_{n}(G)$ in $G$ is bounded in terms of $|C_{G}(F)|$ and $|F|.$ This generalizes a result of Khukhro and Makarenko \cite{k-m} which handles the case $n=1.$