A Combinatorial Proof of a Conjecture of Goldberg and Moon
Canadian mathematical bulletin, Tome 11 (1968) no. 5, pp. 655-661
Voir la notice de l'article provenant de la source Cambridge University Press
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.
Alspach, Brian. A Combinatorial Proof of a Conjecture of Goldberg and Moon. Canadian mathematical bulletin, Tome 11 (1968) no. 5, pp. 655-661. doi: 10.4153/CMB-1968-078-3
@article{10_4153_CMB_1968_078_3,
author = {Alspach, Brian},
title = {A {Combinatorial} {Proof} of a {Conjecture} of {Goldberg} and {Moon}},
journal = {Canadian mathematical bulletin},
pages = {655--661},
year = {1968},
volume = {11},
number = {5},
doi = {10.4153/CMB-1968-078-3},
url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1968-078-3/}
}
[1] 1. Dixon, John D., The maximum order of the group of a tournament. Canad. Math. Bull. 10 (1967) 503-505. Google Scholar
[2] 2. Goldberg, M. and Moon, J.W., On the maximum order of the group of a tournament. Canad. Math. Bull. 9 (1966) 563-569. Google Scholar
[3] 3. Wielandt, H., Finite permutation groups. (Transi. R. Bercov). (Academic Press, New York, 1964). Google Scholar
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