The influence of SΦ−supplemented subgroups on the structure of finite groups

A subgroup H of a finite group G is said to be SΦ−supplemented in G if there exists a subnormal subgroup T of G such that G = HT and H∩T≤Φ(H), where Φ(H) is the Frattini subgroup of H. In this article, we investigate the structure of a finite group G under the assumption that certain subgroups of G are SΦ−supplemented in G. We obtain that a finite group G is nilpotent if and only if every Sylow subgroup of G is SΦ−supplemented in G. And a group G is nilpotent if every maximal subgroup of G is SΦ−supplemented in G. Moreover, the commutativity of G is characterized by using that the minimal subgroups of two non–conjugate maximal subgroups of G are SΦ−supplemented in G.