2019
Том 71
№ 6

All Issues

Zhao Tao

Articles: 2
Article (Ukrainian)

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

Zhang Xirong, Zhao Tao

↓ Abstract   |   Full text (.pdf)

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.

Article (English)

$S\Phi$-Supplemented subgroups of finite groups

Li Xianhua, Zhao Tao

↓ Abstract   |   Full text (.pdf)

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.