On oriented $m$-semiregular representations of finite groups about valency three
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| Format: | Preprint |
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2025
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| _version_ | 1866911067105394688 |
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| author | Xu, Songnian Wong, Dein Zhang, Chi Zhao, Jinxing |
| author_facet | Xu, Songnian Wong, Dein Zhang, Chi Zhao, Jinxing |
| contents | Let $G$ be a group and $m$ a positive integer. We say an $m$-Cayley digraph $Γ$ over $G$ is a digraph that admits a group of automorphisms isomorphic to $G$ acting semiregularly on the vertex set with $m$ orbits. The digraph $Σ$ is $k$-regular if there exists a non-negative integer $k$ such that every vertex has out-valency and in-valency equal to $k$. All digraphs considered in this paper are regular. We say that $G$ admits an oriented $m$-semiregular representation (abbreviated as OmSR) if there exists a regular $m$-Cayley digraph $Γ$ over $G$ such that $Γ$ is oriented and its automorphism group is isomorphic to $G$. In particular, an O1SR is called an ORR. Xia et al. \cite{x2} provided a classification of finite simple groups admitting an ORR of valency 2. Furthermore, in 2022, Du et al. \cite{du2} proved that most finite simple groups admit an OmSR of valency 2 for $m \geq 2$, except for a few exceptional cases. In this paper, we classify the finite groups generated by at most two elements that admit an OmSR of valency 3 for $m \geq 2$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_15405 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On oriented $m$-semiregular representations of finite groups about valency three Xu, Songnian Wong, Dein Zhang, Chi Zhao, Jinxing Group Theory 05C25, 05C20 Let $G$ be a group and $m$ a positive integer. We say an $m$-Cayley digraph $Γ$ over $G$ is a digraph that admits a group of automorphisms isomorphic to $G$ acting semiregularly on the vertex set with $m$ orbits. The digraph $Σ$ is $k$-regular if there exists a non-negative integer $k$ such that every vertex has out-valency and in-valency equal to $k$. All digraphs considered in this paper are regular. We say that $G$ admits an oriented $m$-semiregular representation (abbreviated as OmSR) if there exists a regular $m$-Cayley digraph $Γ$ over $G$ such that $Γ$ is oriented and its automorphism group is isomorphic to $G$. In particular, an O1SR is called an ORR. Xia et al. \cite{x2} provided a classification of finite simple groups admitting an ORR of valency 2. Furthermore, in 2022, Du et al. \cite{du2} proved that most finite simple groups admit an OmSR of valency 2 for $m \geq 2$, except for a few exceptional cases. In this paper, we classify the finite groups generated by at most two elements that admit an OmSR of valency 3 for $m \geq 2$. |
| title | On oriented $m$-semiregular representations of finite groups about valency three |
| topic | Group Theory 05C25, 05C20 |
| url | https://arxiv.org/abs/2507.15405 |