Improving Thompson's Conjecture for Suzuki Groups

Abstract

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 G ≅ M. 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 = 22m+1, then G ≅ Sz(q) × A, where A is abelian. We avoid using classification of finite simple groups in our proofs. © 2016, Copyright © Taylor & Francis Group, LLC.