A note on tournament \(m\)-semiregular representations of finite groups (Q6637138)

A tournament is a digraph \(\Gamma\) such that, for every two distinct vertices \(u, v \in V(\Gamma)\), exactly one of \((u,v)\) and \((v,u)\) is an arc. A tournament has no automorphism \(\alpha\) of order \(2\), since otherwise, there must exist two vertices \(u \not =v\) swapped by \(\alpha\), leading to a contradiction that both \((u,v)\) and \((v,u)\) are arcs of the tournament.\N\NFor a positive integer \(m\), a group \(G\) is said to admit a tournament \(m\)-semiregular representation (\textsf{T}\(m\)\textsf{SR}) if there exists a tournament \(\Gamma\) such that \(\mathrm{Aut}(\Gamma)\) is isomorphic to \(G\) and acts semiregularly on the vertex set of \(\Gamma\) with \(m\) orbits. For \(m=1\) every finite group of odd order admits a \textsf{T}\(m\)\textsf{SR}, with the exceptions of \(\mathbb{Z}_{3}^{2}\) and \(\mathbb{Z}_{3}^{3}\) (see [\textit{L. Babai} and \textit{W. Imrich}, Aequationes Math. 19, 232--244 (1979; Zbl 0422.05034)], \textit{C. D. Godsil} [Aequationes Math. 30, 55--64 (1986; Zbl 0592.05025)]).\N\NThe main result in the paper under review is that every finite group of odd order has a \textsf{T}\(m\)\textsf{SR} for every \(m \geq 2\).