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

Abstract

In [1], 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) x A, where A is abelian. Our proof does not depend on the classification of finite simple groups. © de Gruyter 2015.