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

Abstract

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

98 pages; incorporated referee comments in v2; to appear in Mem. Amer. Math. Soc