On the structure of finite groups isospectral to finite simple groups

Abstract

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 3 D 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.