Finite groups with given weakly $\tau_{\sigma}$-quasinormal subgroups

Authors : Muhammad Tanveer HUSSAİN, Chenchen CAO, Li ZHANG

Pages : 1706-1717

Doi:10.15672/hujms.573548

View : 22 | Download : 8

Publication Date : 2020-10-06

Article Type : Research Paper

Abstract :Let $\sigma=\{{\sigma_i|i\in I}\}$ be a partition of the set of all primes $\mathbb{P}$ and $G$ a finite group. A set $\mathcal{H} $ of subgroups of $G$ is said to be a complete Hall $\sigma$-set of $G$ if every non-identity member of $\mathcal{H}$ is a Hall $\sigma_i$-subgroup of $G$ for some $i\in I$ and $\mathcal{H}$ contains exactly one Hall $\sigma_i$-subgroup of $G$ for every $i$ such that $\sigma_i\cap \piinsert ignore into journalissuearticles values(G);\neq \emptyset$. Let $\tau_{\mathcal{H}}insert ignore into journalissuearticles values(A);=\{ \sigma_{i}\in \sigmainsert ignore into journalissuearticles values(G);\backslash \sigmainsert ignore into journalissuearticles values(A); \ |\ \sigmainsert ignore into journalissuearticles values(A); \cap \sigmainsert ignore into journalissuearticles values(H^{G});\neq\emptyset$ for a Hall $\sigma_{i}$-subgroup $H\in \mathcal{H}\}$. A subgroup $A$ of $G$ is said to be $\tau_{\sigma}$-permutable or $\tau_{\sigma}$-quasinormal in $G$ with respect to $\mathcal{H}$ if $AH^{x}=H^{x}A$ for all $x\in G$ and $H\in \mathcal{H}$ such that $\sigmainsert ignore into journalissuearticles values(H);\subseteq \tau_{\mathcal{H}}insert ignore into journalissuearticles values(A);$, and $\tau_{\sigma}$-permutable or $\tau_{\sigma}$-quasinormal in $G$ if $A$ is $\tau_{\sigma}$-permutable in $G$ with respect to some complete Hall $\sigma$-set of $G$. We say that a subgroup $A$ of $G$ is weakly $\tau_{\sigma}$-quasinormal in $G$ if $G$ has a $\sigma$-subnormal subgroup $T$ such that $AT=G$ and $A\cap T\leq A_{\tau_{\sigma}G}$, where $A_{\tau_{\sigma}G}$ is the subgroup generated by all those subgroups of $A$ which are $\tau_{\sigma}$-quasinormal in $G$. We study the structure of $G$ being based on the assumption that some subgroups of $G$ are weakly $\tau_{\sigma}$-quasinormal in $G$.
Keywords : finite groups, sigma permutable subgroup, au sigma quasinormal subgroup, weakly au sigma quasinormal subgroup, supersoluble group