The structure of a finite group and the maximum $π$-index of its elements
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866910470275858432 |
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| author | Liu, A-Ming Vasil'ev, Andrey V. |
| author_facet | Liu, A-Ming Vasil'ev, Andrey V. |
| contents | Given a set of primes $π$, the $π$-index of an element $x$ of a finite group $G$ is the $π$-part of the index of the centralizer of $x$ in $G$. If $π=\{p\}$ is a singleton, we just say the $p$-index. If the $π$-index of $x$ is equal to $p_1^{k_1}\ldots p^{k_s}$, where $p_1,\ldots,p_s$ are distinct primes, then we set $\exp_π(x)=k_1+\ldots+k_s$. In this short note, we study how the number $ε_π(G)=\max\{ε_π(x):x\in G\}$ restricts the structure of the factor group $G/Z(G)$ of $G$ by its center. First, for a finite group $G$, we prove that $ϕ_p(G/Z(G))\leqε_p(G)$, where $ϕ_p(G/Z(G))$ is the Frattini length of a Sylow $p$-subgroup of $G/Z(G)$. Second, for a $π$-separable finite group $G$, we prove that $l_π(G/Z(G))\leqε_π(G)$, where $l_π(G/Z(G))$ is the $π$-length of $G/Z(G)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_18678 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The structure of a finite group and the maximum $π$-index of its elements Liu, A-Ming Vasil'ev, Andrey V. Group Theory 20E45, 20D20 Given a set of primes $π$, the $π$-index of an element $x$ of a finite group $G$ is the $π$-part of the index of the centralizer of $x$ in $G$. If $π=\{p\}$ is a singleton, we just say the $p$-index. If the $π$-index of $x$ is equal to $p_1^{k_1}\ldots p^{k_s}$, where $p_1,\ldots,p_s$ are distinct primes, then we set $\exp_π(x)=k_1+\ldots+k_s$. In this short note, we study how the number $ε_π(G)=\max\{ε_π(x):x\in G\}$ restricts the structure of the factor group $G/Z(G)$ of $G$ by its center. First, for a finite group $G$, we prove that $ϕ_p(G/Z(G))\leqε_p(G)$, where $ϕ_p(G/Z(G))$ is the Frattini length of a Sylow $p$-subgroup of $G/Z(G)$. Second, for a $π$-separable finite group $G$, we prove that $l_π(G/Z(G))\leqε_π(G)$, where $l_π(G/Z(G))$ is the $π$-length of $G/Z(G)$. |
| title | The structure of a finite group and the maximum $π$-index of its elements |
| topic | Group Theory 20E45, 20D20 |
| url | https://arxiv.org/abs/2405.18678 |