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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2501.13406 |
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| _version_ | 1866916602417512448 |
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| author | Wong, Dein Xu, Songnian Zhang, Chi Zhao, Jinxing |
| author_facet | Wong, Dein Xu, Songnian Zhang, Chi Zhao, Jinxing |
| contents | For a positive integer $m$, a finite group $G$ is said to admit a tournament $m$-semiregular representation (TmSR for short) if there exists a tournament $Γ$ such that the automorphism group of $Γ$ is isomorphic to $G$ and acts semiregularly on the vertex set of $Γ$ with $m$ orbits. Clearly, every finite group of even order does not admit a TmSR for any positive integer $m$, and T1SR is the well-known tournament regular representation (TRR for short). In 1986, Godsil \cite{god} proved, by a probabilistic approach, that the only finite groups of odd order without a TRR are $\mathbb{Z}_3^2$ and $\mathbb{Z}_3^3$ .
More recently, Du \cite{du} proved that every finite group of odd order has a TmSR for every $m \geq 2$. The author of \cite{du} observed that a finite group of odd order has no regular TmSR when $m$ is an even integer, a group of order $1$ has no regular T3SR, and $\mathbb{Z}_3^2$ admits a regular T3SR. At the end of \cite{du}, Du proposed the following problem.
\noindent{\sf\it Problem.} \ \ {\it For every odd integer $m\geq 3$, classify finite groups of odd order which have a regular TmSR.}
The motivation of this paper is to give an answer for the above problem. We proved that if $G$ is a finite group with odd order $n>1$, then $G$ admits a regular TmSR for any odd integer $m\geq 3$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_13406 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Finite groups admitting a regular tournament $m$-semiregular representation Wong, Dein Xu, Songnian Zhang, Chi Zhao, Jinxing Group Theory 05C25 For a positive integer $m$, a finite group $G$ is said to admit a tournament $m$-semiregular representation (TmSR for short) if there exists a tournament $Γ$ such that the automorphism group of $Γ$ is isomorphic to $G$ and acts semiregularly on the vertex set of $Γ$ with $m$ orbits. Clearly, every finite group of even order does not admit a TmSR for any positive integer $m$, and T1SR is the well-known tournament regular representation (TRR for short). In 1986, Godsil \cite{god} proved, by a probabilistic approach, that the only finite groups of odd order without a TRR are $\mathbb{Z}_3^2$ and $\mathbb{Z}_3^3$ . More recently, Du \cite{du} proved that every finite group of odd order has a TmSR for every $m \geq 2$. The author of \cite{du} observed that a finite group of odd order has no regular TmSR when $m$ is an even integer, a group of order $1$ has no regular T3SR, and $\mathbb{Z}_3^2$ admits a regular T3SR. At the end of \cite{du}, Du proposed the following problem. \noindent{\sf\it Problem.} \ \ {\it For every odd integer $m\geq 3$, classify finite groups of odd order which have a regular TmSR.} The motivation of this paper is to give an answer for the above problem. We proved that if $G$ is a finite group with odd order $n>1$, then $G$ admits a regular TmSR for any odd integer $m\geq 3$. |
| title | Finite groups admitting a regular tournament $m$-semiregular representation |
| topic | Group Theory 05C25 |
| url | https://arxiv.org/abs/2501.13406 |