A contribution to the theory of $σ$-properties of a finite group
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866929297744199680 |
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| author | Liu, A-Ming Guo, Wenbin Safonov, Vasily G. Skiba, Alexander N. |
| author_facet | Liu, A-Ming Guo, Wenbin Safonov, Vasily G. Skiba, Alexander N. |
| contents | We characterize some classes of finite soluble groups. In particular, we prove that: a finite group $G$ is supersoluble if and only if $G$ has a normal subgroup $D$ such that $G/D$ is supersoluble and $D$ avoids every chief factor of $G$ between $V^{G}$ and $V_{G}$ for every maximal subgroup $V$ of the generalized Fitting subgroup $F^{*}(G)$ of $G$; a finite soluble group $G$ is a $PST$-group (that is, Sylow permutability is a transitive relation on $G$) if and only if $G$ has a normal subgroup $D$ such that $G/D$ is nilpotent and $D$ avoids every chief factor of $G$ between $V^{G}$ and $V_{G}$ for every subnormal subgroup $A$ of $G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_00004 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A contribution to the theory of $σ$-properties of a finite group Liu, A-Ming Guo, Wenbin Safonov, Vasily G. Skiba, Alexander N. Group Theory 20D10, 20D15, 20D30 We characterize some classes of finite soluble groups. In particular, we prove that: a finite group $G$ is supersoluble if and only if $G$ has a normal subgroup $D$ such that $G/D$ is supersoluble and $D$ avoids every chief factor of $G$ between $V^{G}$ and $V_{G}$ for every maximal subgroup $V$ of the generalized Fitting subgroup $F^{*}(G)$ of $G$; a finite soluble group $G$ is a $PST$-group (that is, Sylow permutability is a transitive relation on $G$) if and only if $G$ has a normal subgroup $D$ such that $G/D$ is nilpotent and $D$ avoids every chief factor of $G$ between $V^{G}$ and $V_{G}$ for every subnormal subgroup $A$ of $G$. |
| title | A contribution to the theory of $σ$-properties of a finite group |
| topic | Group Theory 20D10, 20D15, 20D30 |
| url | https://arxiv.org/abs/2404.00004 |