Recognizing by Spectrum for the Automorphism Groups of Sporadic Simple Groups

TL;DR

This paper proves that the automorphism groups of most sporadic simple groups are uniquely determined by their spectra, with a specific exception involving the automorphism group of J2.

Contribution

It completes the recognition of automorphism groups of all sporadic simple groups by spectrum, except J2, and characterizes the structure of groups isospectral to Aut(J2).

Findings

Automorphism groups of M^cL, M12, M22, He, Suz, O'N are recognizable by spectrum.

The recognizability of Aut(J2) is characterized, with isospectral groups being either Aut(J2) or an extension of a 2-group by A8.

Abstract

The spectrum of a finite group is the set of its element orders, and two groups are said to be isospectral if they have the same spectra. A finite group $G$ is said to be recognizable by spectrum, if every finite group isospectral with $G$ is isomorphic to $G$. We prove that if $S$ is any of the sporadic simple groups $M^cL$, $M_{12}$, $M_{22}$, $He$, $Suz$, $O'N$, then ${\rm Aut}(S)$ is recognizable by spectrum. This finishes the proof of the recognizability by spectrum of the automorphism groups of all sporadic simple groups, except $J_2$. Furthermore, we show that if $G$ is isospectral with ${\rm Aut}(J_2)$, then either $G$ is isomorphic to ${\rm Aut}(J_2)$, or $G$ is an extension of a $2$-group by $\mathbb{A}_8$.