Improving Thompson's Conjecture for Suzuki Groups

Let G be a finite group and cs(G) be the set of conjugacy class sizes of G. In 1987, J. G. Thompson conjectured that, if G is a finite group with Z(G)=1 and M is a nonabelian simple group satisfying that cs(G)=cs(M), then GM. This conjecture has been proved for Suzuki groups in [5]. In this article, we improve this result by proving that, if G is a finite group such that cs(G)=cs(Sz(q)), for q=2(2m+1), then GSz(q)xA, where A is abelian. We avoid using classification of finite simple groups in our proofs.