Finite groups admitting a regular tournament $m$-semiregular representation

TL;DR

This paper proves that all finite groups of odd order greater than one admit a regular tournament m-semiregular representation for any odd integer m ≥ 3, extending previous results and solving an open classification problem.

Contribution

It establishes that every finite group of odd order admits a regular TmSR for all odd m ≥ 3, providing a complete classification for this case.

Findings

All finite groups of odd order > 1 admit a regular TmSR for any odd m ≥ 3.

The result extends previous partial classifications and confirms the conjecture for odd order groups.

The paper solves an open problem posed by Du regarding the classification of such groups.

Abstract

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 $\Gamma$ such that the automorphism group of $\Gamma$ is isomorphic to $G$ and acts semiregularly on the vertex set of $\Gamma$ 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 $m\geq 3$.