Xianhua Li
School Math. Sci., Soochow Univ., Suzhou, China
Tao Zhao
School Sci., Shandong Univ. Technology, Zibo, China)
Abstract
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.
