Finite p-groups with a Frobenius group of automorphisms whose kernel is a cyclic p-group

TL;DR

This paper investigates the structure of finite p-groups with a Frobenius automorphism group, establishing bounds on subgroups' nilpotency class and order based on automorphism properties using Lie ring methods.

Contribution

It introduces new bounds on characteristic subgroups' nilpotency class and order in finite p-groups with Frobenius automorphisms, emphasizing the importance of the cyclic kernel condition.

Findings

Existence of characteristic subgroups with bounded index and nilpotency class

Bounds on subgroup order and rank in terms of automorphism fixed points

The cyclic nature of the kernel is essential for the results

Abstract

Suppose that a finite $p$-group $P$ admits a Frobenius group of automorphisms $FH$ with kernel $F$ that is a cyclic $p$-group and with complement $H$. It is proved that if the fixed-point subgroup $C_P(H)$ of the complement is nilpotent of class $c$, then $P$ has a characteristic subgroup of index bounded in terms of $c$, $|C_P(F)|$, and $|F|$ whose nilpotency class is bounded in terms of $c$ and $|H|$ only. Examples show that the condition of $F$ being cyclic is essential. The proof is based on a Lie ring method and a theorem of the authors and P. Shumyatsky about Lie rings with a metacyclic Frobenius group of automorphisms $FH$. It is also proved that $P$ has a characteristic subgroup of $(|C_P(F)|, |F|)$-bounded index whose order and rank are bounded in terms of $|H|$ and the order and rank of $C_P(H)$, respectively, and whose exponent is bounded in terms of the exponent of $C_P(H)$.