$SΦ$-Supplemented subgroups of finite groups - Li, Xianhua; Zhao, Tao (Ukrains’kyi Matematychnyi Zhurnal, 2012)
We call $H$ an $SΦ$-supplemented subgroup of a finite group $G$ if there exists a subnormal subgroup $T$ of $G$ such that $G = HT$ and $H \bigcap T ≤ Φ(H)$, where $Φ(Н)$ is the Frattini subgroup of $H$. In this paper, we characterize the $p$-nilpotency and supersolubility of a finite group $G$ under the assumption that every subgroup of a Sylow $p$-subgroup of $G$ with given order is $SΦ$-supplemented in $G$. Some results about formations are also obtained.