A note on Frobenius quotient for prime-power divisor of the exponent of finite groups
Fuente:
arXiv
Guardado en:
| Autores principales: | , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2025
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866909486067744768 |
|---|---|
| author | Shi, Jiangtao Liu, Wenjing |
| author_facet | Shi, Jiangtao Liu, Wenjing |
| contents | Let $G$ be a finite group and $n$ be any prime-power divisor of ${\rm exp}(G)$, the exponent of $G$. Frobenius' theorem indicates that $|\{g\in G\mid g^n=1\}|=f_n\cdot n$ for some positive integer $f_n$. We call $f_n$ a Frobenius quotient of $G$ for $n$. Let $\mathcal{F}_{pp}(G)=\{f_n\mid n$ is any prime-power divisor of ${\rm exp}(G)$$\}$ and ${\rm mf}_{pp}(G)$ be the maximum Frobenius quotient in $\mathcal{F}_{pp}(G)$. In this paper, we provide a complete classification of finite group $G$ with ${\rm mf}_{pp}(G)\leq q$, where $q$ is the smallest prime divisor of $|G|$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_06400 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A note on Frobenius quotient for prime-power divisor of the exponent of finite groups Shi, Jiangtao Liu, Wenjing Group Theory 20D10 Let $G$ be a finite group and $n$ be any prime-power divisor of ${\rm exp}(G)$, the exponent of $G$. Frobenius' theorem indicates that $|\{g\in G\mid g^n=1\}|=f_n\cdot n$ for some positive integer $f_n$. We call $f_n$ a Frobenius quotient of $G$ for $n$. Let $\mathcal{F}_{pp}(G)=\{f_n\mid n$ is any prime-power divisor of ${\rm exp}(G)$$\}$ and ${\rm mf}_{pp}(G)$ be the maximum Frobenius quotient in $\mathcal{F}_{pp}(G)$. In this paper, we provide a complete classification of finite group $G$ with ${\rm mf}_{pp}(G)\leq q$, where $q$ is the smallest prime divisor of $|G|$. |
| title | A note on Frobenius quotient for prime-power divisor of the exponent of finite groups |
| topic | Group Theory 20D10 |
| url | https://arxiv.org/abs/2502.06400 |