On finite groups isospectral to simple classical groups

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The spectrum ω(G) of a finite group G is the set of element orders of G. Finite groups G and H are isospectral if their spectra coincide. Suppose that L is a simple classical group of sufficiently large dimension (the lower bound varies for different types of groups but is at most 62) defined over a finite field of characteristic p. It is proved that a finite group G isospectral to L cannot have a nonabelian composition factor which is a group of Lie type defined over a field of characteristic distinct from p. Together with a series of previous results this implies that every finite group G isospectral to L is `close' to L. Namely, if L is a linear or unitary group, then L G(L), in particular, there are only finitely many such groups G for given L. If L is a symplectic or orthogonal group, then G has a unique nonabelian composition factor S and, for given L, there are at most 3 variants for S (including S≃ L).