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Dergiler / Hacettepe Journal of Mathematics and Statistics / 2026 / Volume: 49 Issue: 5

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

Muhammad Tanveer Hussain(University of Science and Technology of China)0000-0003-0884-3056
Chenchen Cao(University of Science and Technology of China)0000-0003-3891-9840
Li Zhang(Anhui Jianzhu University,)0000-0002-3132-744X
Sayfa
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Yayın Tarihi
2020-10-06
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We study the structure of $G$ being based on the assumption that some subgroups of $G$ are weakly $\tau_{\sigma}$-quasinormal in $G$.

📌 3 Maddede Bu Makale

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Amaç & Hipotez:Let $\sigma=\{{\sigma_i|i\in I}\}$ be a partition of the set of all primes $\mathbb{P}$ and $G$ a finite group.
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Temel Bulgular & Veri: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 $G$.
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Sonuç & Katkı:We study the structure of $G$ being based on the assumption that some subgroups of $G$ are weakly $\tau_{\sigma}$-quasinormal in $G$.
🔬Tür:Research Article
📍Kapsam:Genişletilmiş Akademik Örneklem
⚙️Değişkenler:finite groups · $\sigma$-permutable subgroup · $\tau_{\sigma}$-quasinormal subgroup

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 $\sigma_i\cap \pi(G)\neq \emptyset$. 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 $G$.

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Bu Makaleye Atıf Yap

Hussain, M. T., Cao, C., Zhang, L. (2020). Finite groups with given weakly $\tau_{\sigma}$-quasinormal subgroups. Hacettepe Journal of Mathematics and Statistics, Volume: 49 Issue: 5. https://doi.org/10.15672/hujms.573548

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