Supersolvable Frobenius groups with nilpotent centralizers

TL;DR

This paper proves that a finite group with a supersolvable Frobenius automorphism group, acting with trivial centralizer on the group and a nilpotent centralizer for the complement, is itself nilpotent with bounded class.

Contribution

It establishes a bound on the nilpotency class of the group based on the properties of the Frobenius automorphism group and its action.

Findings

G is nilpotent of bounded class

Bound depends on the nilpotency class c and size of FH

Automorphism group conditions influence group structure

Abstract

Let $FH$ be a supersolvable Frobenius group with kernel $F$ and complement $H$. Suppose that a finite group $G$ admits $FH$ as a group of automorphisms in such a manner that $C_G(F)=1$ and $C_{G}(H)$ is nilpotent of class $c$. We show that $G$ is nilpotent of $(c,\left|FH\right|)$-bounded class.