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.
