Vol. 30, No. 1, 1969

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Uniqueness of the graph of a rank three group

David Bertram Wales

Vol. 30 (1969), No. 1, 271–276

Let G be a transitive permutation group on a set Ω. Let {a} be an element of Ω and Δ(a) an orbit of the stabilizer Ga of {a} in G. A graph on the points of Ω can be constructed by joining {a} to Δ(a) and joining each image of {a} by an element of G to the corresponding image of Δ(a). The permutation group G is said to be of rank 3 if Ga has three orbits including {a}. In this paper, conditions are discussed which ensure that the graph of a rank three permutation group is unique in the following sense. Suppose G is a rank three permutation group and Ga has orbits {a},Δ(a), and Γ(a). Is the graph uniquely determined once the permutation representation of Ga on Δ(a) and on Γ(a) is known ? Conditions involving orbit lengths of Ga acting on Δ(a) and on Γ(a) are found which ensure the graph is uniquely determined. The conditions specify how the graph is to be drawn. They give, therefore, a prescription for constructing rank three groups. They shed no light however on the difficult problem of the transitivity of the group of automorphisms of the graph.

Mathematical Subject Classification
Primary: 20.20
Received: 10 January 1969
Published: 1 July 1969
David Bertram Wales