FINITE GROUPS WITH THE SAME CONJUGACY CLASS SIZES AS A FINITE SIMPLE GROUP

Abstract. For a nite group H, let cs(H) denote the set of non-trivial conjugacy class sizes of H and OC(H) be the set of the order components of H. In this paper, we show that if S is a nite simple group with the disconnected prime graph and G is a nite group such that cs(S) = cs(G), then jSj = jG=Z(G)j and OC(S) = OC(G=Z(G)). In particular, we show that for some nite simple group S, G=S Z(G).