On a problem of B. Hartley about a small centralizer in finite and locally finite groups

TL;DR

This paper proves that finite groups with automorphisms having a limited number of fixed points contain soluble subgroups with bounded index and Fitting height, and applies this to locally finite groups with finite centralizers.

Contribution

It establishes bounds on soluble subgroups in finite groups with automorphisms and solves Hartley's problem on locally finite groups with finite centralizers.

Findings

Finite groups with automorphisms of order n and m fixed points have soluble subgroups with bounded index and Fitting height.

Locally finite groups with elements having finite centralizers contain finite index subgroups with finite normal series.

The results provide bounds and structural insights into automorphisms and centralizers in finite and locally finite groups.

Abstract

It is proved that if a finite group $G$ has an automorphism of order $n$ with $m$ fixed points, then $G$ has a soluble subgroup whose index and Fitting height are bounded in terms of $m$ and $n$. As a corollary, a problem of B. Hartley is solved in the affirmative: if a locally finite group $G$ has an element with finite centralizer, then $G$ has a subgroup of finite index which has a finite normal series with locally nilpotent factors.