On recognition of simple classical groups with prime graph independence number $4$ by spectrum

TL;DR

This paper proves that finite groups sharing the same element order spectrum as certain simple classical groups with prime graph independence number 4 are almost simple with the same socle, completing a classification in this area.

Contribution

It establishes the recognition-by-spectrum property for specific simple classical groups with prime graph independence number 4, identifying their automorphism groups.

Findings

Finite groups with the same spectrum as specified classical groups are almost simple.

The recognition-by-spectrum problem is fully addressed for these groups.

The result completes the classification for groups with prime graph independence number 4.

Abstract

Let $L$ be one of the finite simple classical groups $L_8(q)$, $U_8(q)$, $O_{10}^+(q)$, $O_{10}^-(q)$ or $O_{12}^+(q)$, with $q$ odd. We prove that every finite group having the same set of element orders as $L$ is an almost simple group with socle isomorphic to $L$. This completes the study of the recognition-by-spectrum problem for simple classical groups whose prime graph independence number is equal to $4$.