On the structure of finite groups isospectral to finite simple groups

TL;DR

This paper advances the understanding of the spectral properties of finite simple groups, proving that large classical groups are uniquely determined by their element orders, supporting a conjecture about their recognizability.

Contribution

It proves that finite simple classical groups of sufficiently large dimension are almost recognizable by spectrum, confirming Mazurov's conjecture with a specific bound of 60.

Findings

Nonabelian composition factors are either isomorphic to the original group or not of the same Lie type and characteristic.

The result applies to symplectic and orthogonal groups of dimension at least 10.

The conjecture holds with a bound of d_0=60.

Abstract

Finite groups are said to be isospectral if they have the same sets of element orders. A finite nonabelian simple group $L$ is said to be almost recognizable by spectrum if every finite group isospectral to $L$ is an almost simple group with socle isomorphic to $L$. It is known that all finite simple sporadic, alternating and exceptional groups of Lie type, except $J_2$, $A_6$, $A_{10}$ and $^3D_4(2)$, are almost recognizable by spectrum. The present paper is the final step in the proof of the following conjecture due to V.D. Mazurov: there exists a positive integer $d_0$ such that every finite simple classical group of dimension larger than $d_0$ is almost recognizable by spectrum. Namely, we prove that a nonabelian composition factor of a~finite group isospectral to a finite simple symplectic or orthogonal group $L$ of dimension at least 10, is either isomorphic to $L$ or not a group of Lie type in the same characteristic as $L$, and combining this result with earlier work, we deduce that Mazurov's conjecture holds with $d_0=60$.