PGL2(q) cannot be determined by its cs

Abstract

For a finite group G, let Z(G) denote the center of G and cs*(G) be the set of non-trivial conjugacy class sizes of G. In this paper, we show that if G is a finite group such that for some odd prime power q ? 4, cs*(G) = cs*(PGL2(q)), then either G ? PGL2(q) X Z(G) or G contains a normal subgroup N and a non-trivial element t ? G such that N ? PSL2(q)X Z(G), t2 ? N and G = N. ?t?. This shows that the almost simple groups cannot be determined by their set of conjugacy class sizes (up to an abelian direct factor).