On supersolubility of finite groups admitting a Frobenius group of automorphisms with fixed-point-free kernel

TL;DR

This paper studies the structure of finite groups with a Frobenius automorphism group, showing conditions under which the group is supersoluble or a Sylow tower group, based on fixed-point subgroup properties.

Contribution

It proves that if the fixed-point subgroup of the automorphism group is supersoluble and another related subgroup is nilpotent, then the entire group is supersoluble, and characterizes Sylow tower groups.

Findings

G is supersoluble if C_G(H) is supersoluble and C_{G'}(H) is nilpotent

G is a Sylow tower group of a certain type under specific conditions

Provides new criteria for supersolubility in groups with Frobenius automorphisms

Abstract

Assume that a finite group $G$ admits a Frobenius group of automorphisms $FH$ with kernel $F$ and complement $H$ such that $C_{G}(F)=1$. In this paper, we investigate this situation and prove that if $C_G(H)$ is supersoluble and $C_{G'}(H)$ is nilpotent, then $G$ is supersoluble. Also, we show that $G$ is a Sylow tower group of a certain type if $C_{G}(H)$ is a Sylow tower group of the same type.