Automorphism orbits and element orders in finite groups: almost-solubility and the Monster

TL;DR

This paper investigates how the structure of a finite group relates to the number of automorphism orbits and element orders, providing bounds on the index of the soluble radical and characterizing the Monster group.

Contribution

It introduces measures for how far a finite group is from being an AT-group and establishes bounds on the group's soluble radical index based on these measures.

Findings

Bounds on |G:Rad(G)| in terms of automorphism orbit and element order measures

Quantitative characterization of the Monster group

Measures for the deviation from AT-groups

Abstract

For a finite group $G$, we denote by $\omega(G)$ the number of $\operatorname{Aut}(G)$-orbits on $G$, and by $\operatorname{o}(G)$ the number of distinct element orders in $G$. In this paper, we are primarily concerned with the two quantities $\mathfrak{d}(G):=\omega(G)-\operatorname{o}(G)$ and $\mathfrak{q}(G):=\omega(G)/\operatorname{o}(G)$, each of which may be viewed as a measure for how far $G$ is from being an AT-group in the sense of Zhang (that is, a group with $\omega(G)=\operatorname{o}(G)$). We show that the index $|G:\operatorname{Rad}(G)|$ of the soluble radical $\operatorname{Rad}(G)$ of $G$ can be bounded from above both by a function in $\mathfrak{d}(G)$ and by a function in $\mathfrak{q}(G)$ and $\operatorname{o}(\operatorname{Rad}(G))$. We also obtain a curious quantitative characterisation of the Fischer-Griess Monster group $\operatorname{M}$.