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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.

On the structure of finite groups isospectral to finite simple groups