Simple tournaments and sharply transitive groups (Q1201246)

A tournament is a complete graph in which every edge is oriented in some direction. Let \(T\) be a tournament with the set \(V(T)\) of vertices and the set \(E(T)\) of edges. The automorphism group \(\text{Aut} T\) is called sharply transitive, if for every pair \(x,y\in V(T)\) there exists \(\varphi\in\text{Aut} T\) such that \(\varphi(x)=y\) and for any \(x\in V(T)\), \(\varphi\in\text{Aut} T\backslash\{1\} \varphi(x)\neq x\). Tournaments on a set \(V\) are in one-to-one correspondence with commutative binary algebras \((V,\cdot)\) which satisfy \(xy\in\{x,y\}\), \(x,y\in V\). A partition \(V=V_ 1\cup\cdots\cup V_ k\) is called a congruence of \(T\) if it is a congruence of the associated algebra. A tournament \(T\) is called simple if there is no nontrivial congruence on \(T\). In the paper it is shown that a tournament with sharply transitive automorphism group is either a linear ordering or simple.