A dual version of Huppert's conjecture on conjugacy class sizes

Abstract

In [J. Algebra 344 (2011), 205–228], a conjecture of J. G. Thompson for PSLn(q) was proved. It was shown that every finite group G with the property Z(G) = 1 and cs(G) = cs(PSLn(q)) is isomorphic to PSLn(q) where cs(G) is the set of conjugacy class sizes of G. In this article we improve this result for PSL2(q). In fact we prove that if cs(G) = cs(PSL2(q)), for q > 3, then G ≅ PSL2(q) × A, where A is abelian. Our proof does not depend on the classification of finite simple groups.