Finite groups with two supersoluble subgroups

Abstract

Abstract Let G be a finite group. In this paper we obtain some sufficient conditions for the supersolubility of G with two supersoluble non-conjugate subgroups H and K of prime index, not necessarily distinct. It is established that the supersoluble residual of such a group coincides with the nilpotent residual of the derived subgroup. We prove that G is supersoluble in the following cases: one of the subgroups H or K is nilpotent; the derived subgroup G ′ {G^{\prime}} of G is nilpotent; | G : H | = q > r = | G : K | {|G:H|=q>r=|G:K|} and H is normal in G. Also the supersolubility of G with two non-conjugate maximal subgroups M and V is obtained in the following cases: all Sylow subgroups of M and of V are seminormal in G; all maximal subgroups of M and of V are seminormal in G.