On $\Phi$-$\tau$-quasinormal subgroups of finite groups

Document Type : Research Paper


1 Institute of Quantum Information Science‎, ‎Shanxi Datong University‎ ‎Datong 037009‎, ‎P.R‎. ‎China.

2 School of Mathematics and Computer‎, ‎University of Datong of Shanxi‎, ‎Datong 037009‎, ‎P.R‎. ‎China.

3 School of Mathematical Sciences‎, ‎University of Science and Technology of China‎, ‎Hefei‎, ‎230026‎, ‎P.R‎. ‎China.

4 School of mathematics and statistics‎, ‎Jiangsu Normal University‎ ‎Xuzhou‎, ‎221116‎, ‎P.R‎. ‎China.


‎Let $\tau$ be a subgroup functor and $H$ a $p$-subgroup of a finite group $G$‎. ‎Let $\bar{G}=G/H_{G}$ and $\bar{H}=H/H_{G}$‎. ‎We say that $H$ is $\Phi$-$\tau$-quasinormal in $G$ if for some $S$-quasinormal subgroup $\bar{T}$ of $\bar{G}$ and some $\tau$-subgroup $\bar{S}$ of $\bar{G}$ contained in $\bar{H}$‎, ‎$\bar{H}\bar{T}$ is $S$-quasinormal in $\bar{G}$ and $\bar{H}\cap\bar{T}\leq \bar{S}\Phi(\bar{H})$‎. ‎In this paper‎, ‎we study the structure of a group $G$ under the condition that some primary subgroups of $G$ are $\Phi$-$\tau$-quasinormal in $G$‎. ‎Some new characterizations about $p$-nilpotency and solubility of finite groups are obtained.


Main Subjects

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