Further Reflections on Thompsonʹs Conjecture,

Let G be a finite group and let N(G) = {n N G has a conjugacy class C, such that C = n}. Professor J. G. Thompson has conjectured that: If G is a finite group with Z(G) = 1 and M a non-abelian simple group satisfying N(G) = N(M), then G M. We have proved previously that: If M is a sporadic simple group or a simple group having its prime graph with at least three prime graph components, then Thompsonʹs conjecture is correct. In this paper, we shall prove: Let G be a finite group with Z(G) = 1 and M = G2(q) or G2(2)′, where q ≥ 2, such that N(G) = N(M). Then G M.