Extremely closed subgroups and a variant on Glauberman's $Z^*$-theorem
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866916317012951040 |
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| author | Tong-Viet, Hung P. |
| author_facet | Tong-Viet, Hung P. |
| contents | Let $G$ be a finite group and let $H$ be a subgroup of $G$. We say that $H$ is extremely closed in $G$ if $\langle H,H^g\rangle\cap N_G(H)=H$ for all $g\in G.$ In this paper, we determine the structure of finite groups with an extremely closed abelian $p$-subgroup for some prime $p$. In particular, we show that if $G$ contains such a subgroup $H$, then $G=N_G(H)O_{p'}(G).$ This is a variant on the celebrated Glauberman's $Z^*$-Theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_06307 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Extremely closed subgroups and a variant on Glauberman's $Z^*$-theorem Tong-Viet, Hung P. Group Theory 20D05, 20D15 Let $G$ be a finite group and let $H$ be a subgroup of $G$. We say that $H$ is extremely closed in $G$ if $\langle H,H^g\rangle\cap N_G(H)=H$ for all $g\in G.$ In this paper, we determine the structure of finite groups with an extremely closed abelian $p$-subgroup for some prime $p$. In particular, we show that if $G$ contains such a subgroup $H$, then $G=N_G(H)O_{p'}(G).$ This is a variant on the celebrated Glauberman's $Z^*$-Theorem. |
| title | Extremely closed subgroups and a variant on Glauberman's $Z^*$-theorem |
| topic | Group Theory 20D05, 20D15 |
| url | https://arxiv.org/abs/2404.06307 |