On finite soluble groups with almost fixed-point-free automorphisms of non-coprime order

TL;DR

This paper investigates the structure of finite soluble groups with automorphisms of prime power order that have limited fixed points, establishing bounds on the group's p-length and Fitting height based on automorphism properties.

Contribution

It provides new bounds on the p-length and Fitting height of finite soluble groups under automorphisms with restricted fixed points, extending understanding of automorphism actions.

Findings

Bound on p-length in terms of automorphism order and fixed points

Bound on Fitting height for soluble groups with specific automorphisms

Automorphism order and fixed points influence group structure constraints

Abstract

It is proved that if a finite $p$-soluble group $G$ admits an automorphism $\varphi$ of order $p^n$ having at most $m$ fixed points on every $\varphi$-invariant elementary abelian $p'$-section of $G$, then the $p$-length of $G$ is bounded above in terms of $p^n$ and $m$; if in addition the group $G$ is soluble, then the Fitting height of $G$ is bounded above in terms of $p^n$ and $m$. It is also proved that if a finite soluble group $G$ admits an automorphism $\psi$ of order $p^aq^b$ for some primes $p,q$, then the Fitting height of $G$ is bounded above in terms of $|\psi |$ and $|C_G(\psi )|$.