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

Let G be a finite transitive group on a set Ω, let α ∈ Ω and let Gα be the stabilizer of the point α in G. In this paper, we are interested in the proportion [equaction presented] that is, the proportion of elements of Ω lying in a suborbit of cardinality at most two. We show that, if this proportion is greater than 5/6, then each element of Ω lies in a suborbit of cardinality at most two 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 onR containing G in their automorphism groups.