Note on even tournaments whose automorphism groups contain regular subgroups (Q1207794)

A tournament \(A\) is defined to be the adjacency matrix of a complete asymmetric digraph. \(A\) is called even if the inner product of any two distinct row vectors of \(A\) is even. An automorphism of \(A\) is a permutation matrix \(P\) such that \(P^ tAP=A\). The multiplicative group \(G(A)\) of all automorphisms of \(A\) is called the automorphism group of \(A\). The main results are the following theorems. Theorem 1. If the order of 2 modulo every prime divisor of \(v\) is singly even, then there exists a tournament of order \(v\) whose automorphism group contains a regular subgroup which is isomorphic to an arbitrarily given group \(G\) of order \(v\). Theorem 2. If the order of 2 modulo every prime divisor of \(v\) is odd, then there exists an even tournament of order \(v\) whose automorphism group contains a regular subgroup which is isomorphic to an arbitrarily given group \(G\) of order \(v\).