On the nilpotency class of finite groups with a Frobenius group of automorphisms

TL;DR

This paper demonstrates that the nilpotency class bound for finite groups with a Frobenius automorphism group depends on the order of the complement, by constructing examples with unbounded class.

Contribution

It constructs a family of finite nilpotent groups with automorphisms showing the bound on nilpotency class cannot be independent of the automorphism group's order.

Findings

The nilpotency class bound depends on the order of the automorphism group.

Constructs examples of groups with arbitrarily large nilpotency class.

Shows the necessity of the bound's dependence on the automorphism group's size.

Abstract

Suppose that a metacyclic Frobenius group $FH$, with kernel $F$ and complement $H$, acts by automorphisms on a finite group $G$, in such a way that $C_G(F)$ is trivial and $C_G(H)$ is nilpotent. It is known that $G$ is nilpotent and its nilpotency class can be bounded in terms of $|H|$ and the nilpotency class of $C_G(H)$. Until now, it was not clear whether the bound could be made independent of the order of $H$. In this article, we construct a family $\mathfrak{G}$ of finite nilpotent groups, of unbounded nilpotency class. Each group in $\mathfrak{G}$ admits a metacyclic Frobenius group of automorphisms such that the centralizer of the kernel is trivial and the centralizer of the complement is abelian. This shows that the dependence of the bound on the order of $H$ is essential.