Finite groups with SS-supplement

Let G be a finite group. A subgroup H of G is said to be SS -quasinormal in G if there is a subgroup K such that G = H K and H S = S H , for all S ∈ Syl( K ), where Syl( K ) denotes the collection of all Sylow subgroups of K . A subgroup H of G is said to be SS -supplemented in G if there is a subgroup K such that G = H K and H ∩ K is SS -quasinormal in G . In this paper, we investigate the SS -supplemented subgroups and strengthen a result of Skiba which gives a positive answer to an open question of Shemetkov.