$S\Phi$-Supplemented subgroups of finite groups
We call $H$ an $S\Phi$-supplemented subgroup of a finite group $G$ if there exists a subnormal subgroup $T$ of $G$ such that $G = HT$ and $H \bigcap T \leq \Phi(H)$, where $\Phi(Н)$ 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 $S\Phi$-supplemented in $G$. Some results about formations are also obtained.
English version (Springer): Ukrainian Mathematical Journal 64 (2012), no. 1, pp 102-109.
Citation Example: Li Xianhua, Zhao Tao $S\Phi$-Supplemented subgroups of finite groups // Ukr. Mat. Zh. - 2012. - 64, № 1. - pp. 92-99.