Finite groups and Lie rings with an automorphism of order $2^n$

TL;DR

This paper investigates the structure of finite groups and Lie rings with automorphisms of order 2^n, showing that under certain fixed-point conditions, these structures contain large soluble subgroups with bounded derived length and index.

Contribution

It establishes bounds on the existence of characteristic soluble subgroups in finite groups and Lie rings with automorphisms of order 2^n, based on fixed-point properties.

Findings

Existence of characteristic soluble subgroups with bounded derived length

Bounds depend on automorphism order and fixed-point subgroup properties

Results apply to both finite groups and Lie rings

Abstract

Suppose that a finite group $G$ admits an automorphism $\varphi $ of order $2^n$ such that the fixed-point subgroup $C_G(\varphi ^{2^{n-1}})$ of the involution $\varphi ^{2^{n-1}}$ is nilpotent of class $c$. Let $m=|C_G(\varphi)|$ be the number of fixed points of $\varphi$. It is proved that $G$ has a characteristic soluble subgroup of derived length bounded in terms of $n,c$ whose index is bounded in terms of $m,n,c$. A similar result is also proved for Lie rings.