Primitive permutation groups and a characterization of the odd graphs

Suppose that G is a simply transitive primitive permutation group on a finite set Ω such that for α in Ω the stabilizer Gα is 2-transitive on one of its orbits Γ(α) in Ω of length v>2. Peter J. Cameron showed that Gα has another orbit related to Γ(α) of length v(v−1)/l≥2v. Here we show that under certain extra, fairly natural, conditions (including the assumptions that Gα is not faithful on Γ(α)), Gα has a third orbit of length greater than v(v−1), and obtain strong restrictions on the length of this orbit when Gα is more than 2-transitive on Γ(α). In fact if GαΓ(α) Av and Gα is not faithful on Γ(α) we show that either the third orbit has length v(v−1)2, or G is S2v−1 or A2v−1 acting as an automorphism group of the odd graph 0v, where the set of points is identified with Ω and the set of points adjacent to α is precisely Γ(α).

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