On Thompson's conjecture

Abstract

Let G be a finite group and N(G) = {n ∈ N|G has a conjugacy class C, such that |C| = n}. Professor J. G. Thompson has conjectured that "If G be a finite group with Z(G) = 1 and M a nonabelian simple group satisfying that N(G) = N(M), then G ≅ M." We have proved that if M is a sporadic simple group, then Thompson's conjecture is correct. In this paper, we shall further prove that if M is a finite simple group having at least three prime graph components, then the conjecture is also correct. © 1996 Academic Press, Inc.