On the structure of finite groups isospectral to finite simple groups

Mariya A. Grechkoseeva, Andrey V. Vasil'ev

Published 2014-09-29Version 1

Finite groups are said to be isospectral if they have the same sets of element orders. The present paper is the final step in the proof of the following conjecture due to V.D. Mazurov: for every finite nonabelian simple group $L$, apart from a finite number of sporadic, alternating and exceptional groups and apart from several series of classical groups of small dimensions, if a finite group $G$ is isospectral to $L$ then $G$ is an almost simple group with socle isomorphic to $L$. 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 $L$.