On the Length of a Maximal Subgroup of a Finite Group
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910899922534400 |
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| author | Murashka, Viachaslau I. Vasil'ev, Alexander F. |
| author_facet | Murashka, Viachaslau I. Vasil'ev, Alexander F. |
| contents | For a finite group $G$ and its maximal subgroup $M$ we proved that the generalized Fitting height of $M$ can't be less by 2 than the generalized Fitting height of $G$ and the non-$p$-soluble length of $M$ can't be less by 1 than the non-$p$-soluble length of $G$. We constructed a hereditary saturated formation $\mathfrak{F}$ such that $\{n_σ(G, \mathfrak{F})-n_σ(M, \mathfrak{F})\mid G$ is finite $σ$-soluble and $M$ is a maximal subgroup of $G\}=\mathbb{N}\cup\{0\}$ where $n_σ(G, \mathfrak{F})$ denotes the $σ$-nilpotent length of the $\mathfrak{F}$-residual of $G$. This construction shows the results about the generalized lengths of maximal subgroups published in Math. Nachr. (1994) and Mathematics (2020) are not correct. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_24335 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Length of a Maximal Subgroup of a Finite Group Murashka, Viachaslau I. Vasil'ev, Alexander F. Group Theory For a finite group $G$ and its maximal subgroup $M$ we proved that the generalized Fitting height of $M$ can't be less by 2 than the generalized Fitting height of $G$ and the non-$p$-soluble length of $M$ can't be less by 1 than the non-$p$-soluble length of $G$. We constructed a hereditary saturated formation $\mathfrak{F}$ such that $\{n_σ(G, \mathfrak{F})-n_σ(M, \mathfrak{F})\mid G$ is finite $σ$-soluble and $M$ is a maximal subgroup of $G\}=\mathbb{N}\cup\{0\}$ where $n_σ(G, \mathfrak{F})$ denotes the $σ$-nilpotent length of the $\mathfrak{F}$-residual of $G$. This construction shows the results about the generalized lengths of maximal subgroups published in Math. Nachr. (1994) and Mathematics (2020) are not correct. |
| title | On the Length of a Maximal Subgroup of a Finite Group |
| topic | Group Theory |
| url | https://arxiv.org/abs/2503.24335 |