On Thompsonʼs conjecture for some finite simple groups

In this article, we prove a conjecture of J.G. Thompson for an infinite class of simple groups of Lie type. More precisely, we show that every finite group G with the property Z(G)=1 and cs(G)=cs(PSLn(q)) is necessarily isomorphic to PSLn(q). Note that cs(G) is the set of lengths of conjugacy classes of G.