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Tournaments With a Given Automorphism Group

Published online by Cambridge University Press:  20 November 2018

J. W. Moon
J. W. Moon*
Affiliation:
University College, London, England
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The set of all adjacency-preserving automorphisms of the vertex set of a graph form a group which is called the (automorphism) group of the graph. In 1938 Frucht (2) showed that every finite group is isomorphic to the group of some graph. Since then Frucht, Izbicki, and Sabidussi have considered various other properties that a graph having a given group may possess. (For pertinent references and definitions not given here see Ore (4).) The object in this paper is to treat by similar methods a corresponding problem for a class of oriented graphs. It will be shown that a finite group is isomorphic to the group of some complete oriented graph if and only if it has an odd number of elements.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 16 , 1964 , pp. 485 - 489
Copyright
Copyright © Canadian Mathematical Society 1964

References

1. Coxeter, H. S. M. and Moser, W. O. J., Generators and relations for discrete groups (Berlin, 1957).Google Scholar
2. Frucht, R., Herstellung von Graphen mit vorgegebener abstrakter Gruppet Compositio Math., 6 (1938), 239–250.Google Scholar
3. Moon, J. W. and L. Moser, Almost all tournaments are irreducible, Can. Math. Bull., 5 (1962), 61–65.Google Scholar
4. Ore, O., Theory of graphs (Providence, 1962).Google Scholar