Finite groups with some subgroups of prime power order satisfying the partial $ Π$-property
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866914868686225408 |
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| author | Qiu, Zhengtian Ballester-Bolinches, Adolfo |
| author_facet | Qiu, Zhengtian Ballester-Bolinches, Adolfo |
| contents | Let $ H $ be a subgroup of a finite group $ G $. We say that $ H $ satisfies the partial $ Π$-property in $ G $ if there exists a $G$-chief series $ \varGamma_{G}: 1 =G_{0} < G_{1} < \cdot\cdot\cdot < G_{n}= G $ of $ G $ such that $ | G / G_{i-1} : N_{G/G_{i-1}} (HG_{i-1}/G_{i-1}\cap G_{i}/G_{i-1})| $ is a $ π(HG_{i-1}/G_{i-1}\cap G_{i}/G_{i-1}) $-number for every $ G $-chief factor $ G_{i}/G_{i-1} $ of $ \varGamma_{G} $, $1\leq i\leq n$. In this paper, we investigate the structure of a finite group $ G $ under the assumption that some subgroups of prime power order satisfy the partial $ Π$-property. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_01254 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Finite groups with some subgroups of prime power order satisfying the partial $ Π$-property Qiu, Zhengtian Ballester-Bolinches, Adolfo Group Theory Let $ H $ be a subgroup of a finite group $ G $. We say that $ H $ satisfies the partial $ Π$-property in $ G $ if there exists a $G$-chief series $ \varGamma_{G}: 1 =G_{0} < G_{1} < \cdot\cdot\cdot < G_{n}= G $ of $ G $ such that $ | G / G_{i-1} : N_{G/G_{i-1}} (HG_{i-1}/G_{i-1}\cap G_{i}/G_{i-1})| $ is a $ π(HG_{i-1}/G_{i-1}\cap G_{i}/G_{i-1}) $-number for every $ G $-chief factor $ G_{i}/G_{i-1} $ of $ \varGamma_{G} $, $1\leq i\leq n$. In this paper, we investigate the structure of a finite group $ G $ under the assumption that some subgroups of prime power order satisfy the partial $ Π$-property. |
| title | Finite groups with some subgroups of prime power order satisfying the partial $ Π$-property |
| topic | Group Theory |
| url | https://arxiv.org/abs/2404.01254 |