On the existence of \(k\)-tournaments with given automorphism group
A \(k\)-tournament \(T\) on \(n\) vertices consists of an \(n\)-set \(V(T)\) of vertices and the \({n\choose k}\)-collection of \(k\)-subsets of \(V(T)\), where each \(k\)-subset has been assigned one of the \(k!\) possible linear orders. The main result of this lucid paper is that, for any finite group \(G\) and any integer \(k\geq 3\), there exists a \(k\)-tournament whose automorphism group is isomorphic to \(G\) if and only if \(|G|\) and \(k\) are relatively prime. This extends a result of \textit{J. W. Moon} [Can. J. Math. 16, 485-489 (1964; Zbl 0121.40204)] that a finite group is isomorphic to the automorphism group of a tournament (i.e., a 2-tournament) if and only if its order is odd.
scientific article; zbMATH DE number 897582