A Combinatorial Proof of a Conjecture of Goldberg and Moon

Abstract

Let Tn denote a tournament of order n, let G(Tn) denote the automorphism group of Tn, let |G| denote the order of the group G, and let g(n) denote the maximum of |G(Tn)| taken over all tournaments Tn of order n. Goldberg and Moon conjectured [2] that for all n≥1 with equality holding if and only if n is a power of 3. In an addendum to [2] it was pointed out that their conjecture is equivalent to the conjecture that if G is any odd order subgroup of Sn, the symmetric group of degree n, then with equality possible if and only if n is a power of 3. The latter conjecture was proved in [1] by John D. Dixon who made use of the Feit-Thompson theorem in his proof. In this paper we avoid use of the Feit-Thompson result and give a combinatorial proof of the Goldberg-Moon conjecture.