# Zhao Tao

### Weakly *SS*-Quasinormal Minimal Subgroups and the Nilpotency of a Finite Group

Ukr. Mat. Zh. - 2014. - 66, № 2. - pp. 187–194

A subgroup *H* is said to be an *s*-permutable subgroup of a finite group *G* provided that the equality *HP* =*PH* holds for every Sylow subgroup *P* of *G.* Moreover, *H* is called *SS*-quasinormal in *G* if there exists a supplement *B* of *H* to *G* such that *H* permutes with every Sylow subgroup of *B.* We show that *H* is weakly *SS*-quasinormal in *G* if there exists a normal subgroup *T* of *G* such that *HT* is *s*-permutable and *H \ T* is *SS*-quasinormal in *G.* We study the influence of some weakly *SS*-quasinormal minimal subgroups on the nilpotency of a finite group *G.* Numerous results known from the literature are unified and generalized.

### $S\Phi$-Supplemented subgroups of finite groups

Ukr. Mat. Zh. - 2012. - 64, № 1. - pp. 92-99

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.