Finite transitive groups having many suborbits of cardinality at most 2 and an application to the enumeration of Cayley graphs

Let G be a finite transitive group on a set omega , let alpha is an element of omega , and let G alpha be the stabilizer of the point alpha in G. In this paper, we are interested in the proportion |{omega is an element of omega|omega lies in a G alpha-orbit of cardinality at most 2}| |omega|,that is, the proportion of elements of omega lying in a suborbit of cardinality at most 2. We show that, if this proportion is greater than 5/6 , then each element of omega lies in a suborbit of cardinality at most 2, and hence G is classified by a result of Bergman and Lenstra. We also classify the permutation groups attaining the bound 5/6 . We use these results to answer a question concerning the enumeration of Cayley graphs. Given a transitive group G containing a regular subgroup R, we determine an upper bound on the number of Cayley graphs on R containing G in their automorphism groups.