Extensions of a theorem of P. Hall on indexes of maximal subgroups
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866913635108913152 |
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| author | Beltrán, Antonio Shao, Changguo |
| author_facet | Beltrán, Antonio Shao, Changguo |
| contents | We extend a classical theorem of P. Hall that claims that if the index of every maximal subgroup of a finite group $G$ is a prime or the square of a prime, then $G$ is solvable. Precisely, we prove that if one allows, in addition, the possibility that every maximal subgroup of $G$ is nilpotent instead of having prime or squared-prime index, then $G$ continues to be solvable. Likewise, we obtain the solvability of $G$ when we assume that every proper non-maximal subgroup of $G$ lies in some subgroup of index prime or squared prime. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_02249 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Extensions of a theorem of P. Hall on indexes of maximal subgroups Beltrán, Antonio Shao, Changguo Group Theory 20E28, 20D15, 20D06 We extend a classical theorem of P. Hall that claims that if the index of every maximal subgroup of a finite group $G$ is a prime or the square of a prime, then $G$ is solvable. Precisely, we prove that if one allows, in addition, the possibility that every maximal subgroup of $G$ is nilpotent instead of having prime or squared-prime index, then $G$ continues to be solvable. Likewise, we obtain the solvability of $G$ when we assume that every proper non-maximal subgroup of $G$ lies in some subgroup of index prime or squared prime. |
| title | Extensions of a theorem of P. Hall on indexes of maximal subgroups |
| topic | Group Theory 20E28, 20D15, 20D06 |
| url | https://arxiv.org/abs/2501.02249 |