The influence of weakly SΦ-supplemented subgroups on the structure of finite groups

Let G be a finite group. A subgroup H of G is said to be s-permutable in G if HP = PH for all Sylow subgroups P of G. A subgroup H of G is said to be weakly S Phi-supplemented in G if G has a subgroup K such that G = HK and H boolean AND K <= Phi(H)H(s)G, where Phi(H) is the Frattini subgroup of H, and H(s)G is the subgroup of H generated by all those subgroups of H which are s-permutable in G. In this paper, we investigate the structure of G under the assumption that certain subgroups of fixed prime power orders are weakly S Phi-supplemented in G. Our main results improve and extend some results in the literature.