Finite groups with some particular maximal invariant subgroups being nilpotent or all non-nilpotent maximal invariant subgroups being normal
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866914897860755456 |
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| author | Shi, Jiangtao Xu, Fanjie |
| author_facet | Shi, Jiangtao Xu, Fanjie |
| contents | Let $A$ and $G$ be finite groups such that $A$ acts coprimely on $G$ by automorphisms. We provide a complete classification of a finite group $G$ in which every maximal $A$-invariant subgroup containing the normalizer of some $A$-invariant Sylow subgroup is nilpotent. Moreover, we show that both the hypothesis that every maximal $A$-invariant subgroup of $G$ containing the normalizer of some $A$-invariant Sylow subgroup is nilpotent and the hypothesis that every non-nilpotent maximal $A$-invariant subgroup of $G$ is normal are equivalent. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_01249 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Finite groups with some particular maximal invariant subgroups being nilpotent or all non-nilpotent maximal invariant subgroups being normal Shi, Jiangtao Xu, Fanjie Group Theory 20D10, 20D25 Let $A$ and $G$ be finite groups such that $A$ acts coprimely on $G$ by automorphisms. We provide a complete classification of a finite group $G$ in which every maximal $A$-invariant subgroup containing the normalizer of some $A$-invariant Sylow subgroup is nilpotent. Moreover, we show that both the hypothesis that every maximal $A$-invariant subgroup of $G$ containing the normalizer of some $A$-invariant Sylow subgroup is nilpotent and the hypothesis that every non-nilpotent maximal $A$-invariant subgroup of $G$ is normal are equivalent. |
| title | Finite groups with some particular maximal invariant subgroups being nilpotent or all non-nilpotent maximal invariant subgroups being normal |
| topic | Group Theory 20D10, 20D25 |
| url | https://arxiv.org/abs/2408.01249 |