$m$-partite oriented semiregular representation of valency 3 for finite groups
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908668855844864 |
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| author | Xu, Songnian Wong, Dein Zhen, Wenhao |
| author_facet | Xu, Songnian Wong, Dein Zhen, Wenhao |
| contents | Let $G$ be a finite group and $m \geq 2$ a positive integer. We say that $G$ admits an \emph{oriented $m$-semiregular representation} (abbreviated as OmSR) if there exists a $m$-Cayley digraph $Γ$ over $G$ such that $Γ$ is oriented and $\mathrm{Aut}(Γ) \cong G$. In \cite{xu1}, we classified finite groups generated by at most two elements that admit an OmSR of valency 3 for $m \geq 2$ and $G \ncong \mathbb{Z}_1$. In this article, we consider $m$-partite digraphs.We say a finite group $G$ admits an \emph{$m$-partite oriented semiregular representation} ($m$-partite digraphical representation), abbreviated as \emph{$m$-POSR} (\emph{$m$-PDR}), if there exists an \emph{oriented} $m$-partite Cayley digraph (\emph{$m$-partite Cayley digraph}) $Γ$ with $\mathrm{Aut}(Γ) \cong G$. In this paper, we classify finite groups generated by at most two elements that admit $m$-POSR. Since if $G$ admits an $m$-POSR, then $G$ must also admit an $m$-PDR (while the converse does not hold), as a natural consequence, we also provide a complete classification for groups $G=\langle x,y\rangle$ that admit $m$-PDR of valency 3. This complements the results in \cite{xu2}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_12660 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $m$-partite oriented semiregular representation of valency 3 for finite groups Xu, Songnian Wong, Dein Zhen, Wenhao Group Theory 05C25 Let $G$ be a finite group and $m \geq 2$ a positive integer. We say that $G$ admits an \emph{oriented $m$-semiregular representation} (abbreviated as OmSR) if there exists a $m$-Cayley digraph $Γ$ over $G$ such that $Γ$ is oriented and $\mathrm{Aut}(Γ) \cong G$. In \cite{xu1}, we classified finite groups generated by at most two elements that admit an OmSR of valency 3 for $m \geq 2$ and $G \ncong \mathbb{Z}_1$. In this article, we consider $m$-partite digraphs.We say a finite group $G$ admits an \emph{$m$-partite oriented semiregular representation} ($m$-partite digraphical representation), abbreviated as \emph{$m$-POSR} (\emph{$m$-PDR}), if there exists an \emph{oriented} $m$-partite Cayley digraph (\emph{$m$-partite Cayley digraph}) $Γ$ with $\mathrm{Aut}(Γ) \cong G$. In this paper, we classify finite groups generated by at most two elements that admit $m$-POSR. Since if $G$ admits an $m$-POSR, then $G$ must also admit an $m$-PDR (while the converse does not hold), as a natural consequence, we also provide a complete classification for groups $G=\langle x,y\rangle$ that admit $m$-PDR of valency 3. This complements the results in \cite{xu2}. |
| title | $m$-partite oriented semiregular representation of valency 3 for finite groups |
| topic | Group Theory 05C25 |
| url | https://arxiv.org/abs/2511.12660 |