Recognizability by spectrum of alternating groups

TL;DR

This paper proves that most simple alternating groups are uniquely determined by their element order spectrum, meaning no other non-isomorphic group shares the same spectrum, except for specific small cases.

Contribution

It establishes the recognizability by spectrum of simple alternating groups for all but two small cases, advancing understanding of group spectra and their uniqueness.

Findings

Most simple alternating groups are recognizable by spectrum.

Groups with the same spectrum as a nonabelian simple group have at most one such factor.

Excluded cases are n=6 and n=10.

Abstract

The spectrum of a group is the set of its element orders. A finite group $G$ is said to be recognizable by spectrum if every finite group that has the same spectrum as $G$ is isomorphic to $G$. We prove that the simple alternating groups $A_n$ are recognizable by spectrum when $n\neq 6, 10$. This implies that every finite group with the same spectrum as that of a finite nonabelian simple group, has at most one nonabelian composition factor