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On the Group of a Directed Graph

Published online by Cambridge University Press:  20 November 2018

Robert L. Hemminger
Robert L. Hemminger*
Affiliation:
Vanderbilt University, Nashville, Tennessee
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In 1938, Frucht (2) proved that for any given finite group G there exists a finite symmetric graph X such that G(X) is abstractly isomorphic to G. Since G(X) is a permutation group, it is natural to ask the following related question : If P is a given finite permutation group, does there exist a symmetric (and more generally a directed) graph X such that G(X) and P are isomorphic (see Convention below) as permutation groups? The answer for the symmetric case is negative as seen in (3) and more recently in (1). It is the purpose of this paper to deal with this problem further, especially in the directed case. In §3, we supplement Kagno's results (3, pp. 516-520) for symmetric graphs by giving the corresponding results for directed graphs.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 18 , 1966 , pp. 211 - 220
Copyright
Copyright © Canadian Mathematical Society 1966

References

1. Chao, Chong-Yun, On a theorem of Sabidussi, Proc. Amer. Math. Soc, 15 (1964), 291–292.Google Scholar
2. Frucht, R., Herstellung von Graphen mit vorgegebener abstrakter Gruppe, Compositio Math., 6(1938), 239–250.Google Scholar
3. Kagno, I. N., Linear graphs of degree ≤ 6 and their groups, Amer. J. Math., 68 (1946), 505–520. Corrections, Amer. J. Math., 77 (1955), 392.Google Scholar
4. Miller, G. A., Memoir on the substitution groups whose degree does not exceed eight, Amer. J. Math., 77 (1899), 288–337.Google Scholar
5. Ore, O., Theory of graphs (Providence, R.I., 1962).Google Scholar
6. Zassenhaus, H., The theory of groups, 2nd ed. (New York, 1958).Google Scholar