Finite groups and Lie rings with a metacyclic Frobenius group of automorphisms

TL;DR

This paper investigates the structure of finite groups and Lie rings admitting a Frobenius group of automorphisms with a cyclic kernel, establishing bounds on nilpotent subgroups and extending previous results with new methods.

Contribution

It generalizes earlier theorems by providing bounds on nilpotent characteristic subgroups in finite groups and Lie rings with metacyclic Frobenius automorphisms, using advanced algebraic techniques.

Findings

Existence of a nilpotent characteristic subgroup with bounded index

Bound on the nilpotency class depending on parameters

Extension of results to Lie rings with similar automorphism groups

Abstract

Suppose that a finite group $G$ admits a Frobenius group of automorphisms FH of coprime order with cyclic kernel F and complement H such that the fixed point subgroup $C_G(H)$ of the complement is nilpotent of class $c$. It is proved that $G$ has a nilpotent characteristic subgroup of index bounded in terms of $c$, $|C_G(F)|$, and $|FH|$ whose nilpotency class is bounded in terms of $c$ and $|H|$ only. This generalizes the previous theorem of the authors and P. Shumyatsky, where for the case of $C_G(F)=1$ the whole group was proved to be nilpotent of $(c,|H|)$-bounded class. Examples show that the condition of $F$ being cyclic is essential. B. Hartley's theorem based on the classification provides reduction to soluble groups. Then representation theory arguments are used to bound the index of the Fitting subgroup. Lie ring methods are used for nilpotent groups. A similar theorem on Lie rings with a metacyclic Frobenius group of automorphisms $FH$ is also proved.