Mao, Y., Ma, X., Tang, X., Huang, J. (2017). On $\Phi$-$\tau$-quasinormal subgroups of finite groups. Bulletin of the Iranian Mathematical Society, 43(7), 2169-2182.

Y. Mao; X. Ma; X. Tang; J. Huang. "On $\Phi$-$\tau$-quasinormal subgroups of finite groups". Bulletin of the Iranian Mathematical Society, 43, 7, 2017, 2169-2182.

Mao, Y., Ma, X., Tang, X., Huang, J. (2017). 'On $\Phi$-$\tau$-quasinormal subgroups of finite groups', Bulletin of the Iranian Mathematical Society, 43(7), pp. 2169-2182.

Mao, Y., Ma, X., Tang, X., Huang, J. On $\Phi$-$\tau$-quasinormal subgroups of finite groups. Bulletin of the Iranian Mathematical Society, 2017; 43(7): 2169-2182.

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

^{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.

Receive Date: 15 January 2016,
Revise Date: 17 December 2016,
Accept Date: 15 January 2017

Abstract

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.