Normalizers of Sylow subgroups in finite reflection groups
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866929488579788800 |
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| author | Townsend, Kane Douglas |
| author_facet | Townsend, Kane Douglas |
| contents | Let $W$ be a finite reflection group, either real or complex, and $S_\ell$ a Sylow $\ell$-subgroup of $W$. We prove the existence of a semidirect product decomposition of $N_W(S_\ell)$ in terms of the unique parabolic subgroup of $W$ minimally containing $S_\ell$ and known decompositions of normalizers of parabolic subgroups. In the real setting, the description follows from the existence of Sylow $\ell$-subgroups stable under the Coxeter diagram automorphisms of finite reflection groups with no proper parabolic subgroup containing a Sylow $\ell$-subgroup. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_04044 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Normalizers of Sylow subgroups in finite reflection groups Townsend, Kane Douglas Group Theory 20F55 (primary), 20D20 (secondary) Let $W$ be a finite reflection group, either real or complex, and $S_\ell$ a Sylow $\ell$-subgroup of $W$. We prove the existence of a semidirect product decomposition of $N_W(S_\ell)$ in terms of the unique parabolic subgroup of $W$ minimally containing $S_\ell$ and known decompositions of normalizers of parabolic subgroups. In the real setting, the description follows from the existence of Sylow $\ell$-subgroups stable under the Coxeter diagram automorphisms of finite reflection groups with no proper parabolic subgroup containing a Sylow $\ell$-subgroup. |
| title | Normalizers of Sylow subgroups in finite reflection groups |
| topic | Group Theory 20F55 (primary), 20D20 (secondary) |
| url | https://arxiv.org/abs/2211.04044 |