Year 2020, Volume 49 , Issue 5, Pages 1706 - 1717 2020-10-06

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

Muhammad Tanveer HUSSAİN [1] , Chenchen CAO [2] , Li ZHANG [3]


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 $\sigma_i\cap \pi(G)\neq \emptyset$. Let $\tau_{\mathcal{H}}(A)=\{ \sigma_{i}\in \sigma(G)\backslash \sigma(A) \ |\ \sigma(A) \cap \sigma(H^{G})\neq\emptyset$ 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 $\sigma(H)\subseteq \tau_{\mathcal{H}}(A)$, 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 $G$.
finite groups, $\sigma$-permutable subgroup, $\tau_{\sigma}$-quasinormal subgroup, weakly $\tau_{\sigma}$-quasinormal subgroup, supersoluble group
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Primary Language en
Subjects Mathematics
Journal Section Mathematics
Authors

Orcid: 0000-0003-0884-3056
Author: Muhammad Tanveer HUSSAİN
Institution: University of Science and Technology of China
Country: China


Orcid: 0000-0003-3891-9840
Author: Chenchen CAO (Primary Author)
Institution: University of Science and Technology of China
Country: China


Orcid: 0000-0002-3132-744X
Author: Li ZHANG
Institution: Anhui Jianzhu University,
Country: China


Supporting Institution NNSF of China
Project Number 11771409
Dates

Publication Date : October 6, 2020

Bibtex @research article { hujms573548, journal = {Hacettepe Journal of Mathematics and Statistics}, issn = {2651-477X}, eissn = {2651-477X}, address = {}, publisher = {Hacettepe University}, year = {2020}, volume = {49}, pages = {1706 - 1717}, doi = {10.15672/hujms.573548}, title = {Finite groups with given weakly \$\\tau\_\{\\sigma\}\$-quasinormal subgroups}, key = {cite}, author = {Hussai̇n, Muhammad Tanveer and Cao, Chenchen and Zhang, Li} }
APA Hussai̇n, M , Cao, C , Zhang, L . (2020). Finite groups with given weakly $\tau_{\sigma}$-quasinormal subgroups . Hacettepe Journal of Mathematics and Statistics , 49 (5) , 1706-1717 . DOI: 10.15672/hujms.573548
MLA Hussai̇n, M , Cao, C , Zhang, L . "Finite groups with given weakly $\tau_{\sigma}$-quasinormal subgroups" . Hacettepe Journal of Mathematics and Statistics 49 (2020 ): 1706-1717 <https://dergipark.org.tr/en/pub/hujms/issue/57199/573548>
Chicago Hussai̇n, M , Cao, C , Zhang, L . "Finite groups with given weakly $\tau_{\sigma}$-quasinormal subgroups". Hacettepe Journal of Mathematics and Statistics 49 (2020 ): 1706-1717
RIS TY - JOUR T1 - Finite groups with given weakly $\tau_{\sigma}$-quasinormal subgroups AU - Muhammad Tanveer Hussai̇n , Chenchen Cao , Li Zhang Y1 - 2020 PY - 2020 N1 - doi: 10.15672/hujms.573548 DO - 10.15672/hujms.573548 T2 - Hacettepe Journal of Mathematics and Statistics JF - Journal JO - JOR SP - 1706 EP - 1717 VL - 49 IS - 5 SN - 2651-477X-2651-477X M3 - doi: 10.15672/hujms.573548 UR - https://doi.org/10.15672/hujms.573548 Y2 - 2019 ER -
EndNote %0 Hacettepe Journal of Mathematics and Statistics Finite groups with given weakly $\tau_{\sigma}$-quasinormal subgroups %A Muhammad Tanveer Hussai̇n , Chenchen Cao , Li Zhang %T Finite groups with given weakly $\tau_{\sigma}$-quasinormal subgroups %D 2020 %J Hacettepe Journal of Mathematics and Statistics %P 2651-477X-2651-477X %V 49 %N 5 %R doi: 10.15672/hujms.573548 %U 10.15672/hujms.573548
ISNAD Hussai̇n, Muhammad Tanveer , Cao, Chenchen , Zhang, Li . "Finite groups with given weakly $\tau_{\sigma}$-quasinormal subgroups". Hacettepe Journal of Mathematics and Statistics 49 / 5 (October 2020): 1706-1717 . https://doi.org/10.15672/hujms.573548
AMA Hussai̇n M , Cao C , Zhang L . Finite groups with given weakly $\tau_{\sigma}$-quasinormal subgroups. Hacettepe Journal of Mathematics and Statistics. 2020; 49(5): 1706-1717.
Vancouver Hussai̇n M , Cao C , Zhang L . Finite groups with given weakly $\tau_{\sigma}$-quasinormal subgroups. Hacettepe Journal of Mathematics and Statistics. 2020; 49(5): 1706-1717.