Simple tournaments and sharply transitive groups

If the relation E is transitive, i.e., if (x, y) E E and (y, z) E E imply that (x, z) E E, we speak of a transitive tournament or a total order. Clearly (Z, <) is a total order. We tend to visualize tournaments by considering every edge (x, y) E E as an arrow leading from x to y. In this sense every tournament can also be considered as a complete graph in which every edge is oriented in some direction. We shall also say that x dominates y if (x, y) E E and that a subset A of V(T) dominates a subset B of V(T) if every element of A dominates every element of B. The automorphism group of a tournament T will be denoted by Aut T. 1 will denote the unit element of this group, i.e., the identity mapping. We say that Aut T is transitive, if there is a # E Aut T to every pair X, y of elements of V(T) such that +x = y. If @x #x for any $ E Aut T\ { l} and any x E V(T) we call Aut T jixed point free. A transitive fixed free group is also called a sharply transitive group. It is easy to see that Aut(Z, <) is transitive and fixed point free.