2025/01/23 by Wong, Dein, Xu, Songnian, Zhang, Chi +1
#05C25 #FOS: Mathematics #Group Theory (math.GR)
paper · doi:10.48550/arxiv.2501.13406
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 \citegod proved, by a probabilistic approach, that the only finite groups of odd order without a TRR are ℤ32 and ℤ33 . More recently, Du \citedu proved that every finite group of odd order has a TmSR for every m ≥ 2. The author of \citedu 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 ℤ32 admits a regular T3SR. At the end of \citedu, Du proposed the following problem. \noindent\sf\it Problem. \it For every odd integer m≥ 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≥ 3.