On generalizations of Iwasawa's theorem
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | |
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| _version_ | 1866929290959912960 |
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| author | Shi, Jiangtao Xu, Fanjie Shan, Mengjiao |
| author_facet | Shi, Jiangtao Xu, Fanjie Shan, Mengjiao |
| contents | Iwasawa's theorem indicates that a finite group $G$ is supersolvable if and only if all maximal chains of the identity in $G$ have the same length. As generalizations of Iwasawa's theorem, we provide some characterizations of the structure of a finite group $G$ in which all maximal chains of every minimal subgroup have the same length. Moreover, let $δ(G)$ be the number of subgroups of $G$ all of whose maximal chains in $G$ do not have the same length, we prove that $G$ is a non-solvable group with $δ(G)\leq 16$ if and only if $G\cong A_5$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_17960 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On generalizations of Iwasawa's theorem Shi, Jiangtao Xu, Fanjie Shan, Mengjiao Group Theory 20D10 Iwasawa's theorem indicates that a finite group $G$ is supersolvable if and only if all maximal chains of the identity in $G$ have the same length. As generalizations of Iwasawa's theorem, we provide some characterizations of the structure of a finite group $G$ in which all maximal chains of every minimal subgroup have the same length. Moreover, let $δ(G)$ be the number of subgroups of $G$ all of whose maximal chains in $G$ do not have the same length, we prove that $G$ is a non-solvable group with $δ(G)\leq 16$ if and only if $G\cong A_5$. |
| title | On generalizations of Iwasawa's theorem |
| topic | Group Theory 20D10 |
| url | https://arxiv.org/abs/2403.17960 |