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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2402.13365 |
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| _version_ | 1866911781178310656 |
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| author | Lewis, Mark L. Shen, Zhencai Yan, Quanfu |
| author_facet | Lewis, Mark L. Shen, Zhencai Yan, Quanfu |
| contents | Let $G$ be a finite group and $N_Ω(G)$ be the intersection of the normalizers of all subgroups belonging to the set $Ω(G),$ where $Ω(G)$ is a set of all subgroups of $G$ which have some theoretical group property. In this paper, we show that $N_Ω(G)= Z_{\infty}(G)$ if $Ω(G)$ is one of the following: (i) the set of all self-normalizing subgroups of $G$; (ii) the set of all subgroups of $G$ satisfying the subnormalizer condition in $G$; (iii) the set of all pronormal subgroups of $G$; (iv) the set of all $\mathscr{H}$-subgroups of $G$; (v) the set of all weakly normal subgroups of $G$; (vi) the set of all $NE$-subgroups of $G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_13365 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Some results on the norm of finite groups Lewis, Mark L. Shen, Zhencai Yan, Quanfu Group Theory Let $G$ be a finite group and $N_Ω(G)$ be the intersection of the normalizers of all subgroups belonging to the set $Ω(G),$ where $Ω(G)$ is a set of all subgroups of $G$ which have some theoretical group property. In this paper, we show that $N_Ω(G)= Z_{\infty}(G)$ if $Ω(G)$ is one of the following: (i) the set of all self-normalizing subgroups of $G$; (ii) the set of all subgroups of $G$ satisfying the subnormalizer condition in $G$; (iii) the set of all pronormal subgroups of $G$; (iv) the set of all $\mathscr{H}$-subgroups of $G$; (v) the set of all weakly normal subgroups of $G$; (vi) the set of all $NE$-subgroups of $G$. |
| title | Some results on the norm of finite groups |
| topic | Group Theory |
| url | https://arxiv.org/abs/2402.13365 |