On SS-quasinormalities of the maximal subgroup series of finite groups
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| Format: | Preprint |
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2026
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| _version_ | 1866908887621304320 |
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| author | Meng, Wei Lu, Jiakuan |
| author_facet | Meng, Wei Lu, Jiakuan |
| contents | Let $G$ be finite group. A subgroup $H$ of $G$ is said to be an $SS$-quasinormal subgroup of $G$, if there exists a subgroup $B$ of $G$ such that $G = HB$ and $H$ permutes with every Sylow subgroup of $B$. Let $Ω: G=G_0>G_1>\cdots>G_{n-1}>G_n=1$ be a maximal subgroup series of $G$, where $G_i$ is a maximal subgroup of $G_{i-1}$ for every $i = 1, \ldots , n$. In this paper, we investigate the finite groups $G$ that admit an $SS$-quasinormal maximal subgroup series, i.e., all $G_i$ are $SS$-quasinormal in $G$. First, we prove that if $G$ possesses an $SS$-quasinormal maximal subgroup series, then $G$ is solvable. Furthermore, we show that $G$ is supersolvable if and only if $G$ possesses an $SS$-quasinormal maximal subgroup series which is subnormal in $G$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_14268 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On SS-quasinormalities of the maximal subgroup series of finite groups Meng, Wei Lu, Jiakuan Group Theory 20D10, 20D30 Let $G$ be finite group. A subgroup $H$ of $G$ is said to be an $SS$-quasinormal subgroup of $G$, if there exists a subgroup $B$ of $G$ such that $G = HB$ and $H$ permutes with every Sylow subgroup of $B$. Let $Ω: G=G_0>G_1>\cdots>G_{n-1}>G_n=1$ be a maximal subgroup series of $G$, where $G_i$ is a maximal subgroup of $G_{i-1}$ for every $i = 1, \ldots , n$. In this paper, we investigate the finite groups $G$ that admit an $SS$-quasinormal maximal subgroup series, i.e., all $G_i$ are $SS$-quasinormal in $G$. First, we prove that if $G$ possesses an $SS$-quasinormal maximal subgroup series, then $G$ is solvable. Furthermore, we show that $G$ is supersolvable if and only if $G$ possesses an $SS$-quasinormal maximal subgroup series which is subnormal in $G$. |
| title | On SS-quasinormalities of the maximal subgroup series of finite groups |
| topic | Group Theory 20D10, 20D30 |
| url | https://arxiv.org/abs/2603.14268 |