On the Group of a Directed Graph

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.