ON THOMPSON’S CONJECTURE FOR Aut(J\(_2\)) AND Aut(McL)

Abstract

Let G be a finite group and N(G) be the set of conjugacy class sizes of G.  In 1980s J. G. Thompson conjectured: If G is a finite group with trivial center and S is a non-abelian finite simple group satisfying that N(G) = N(S), then G \(\cong\) S.  Here, we generalize the conjecture to the automorphism groups of J₂ and McL.  As a corollary this result extends the conjecture to all almost sporadic simple groups.