Let G ≤ Sym(Ω) be a transitive permutation group on a finite set Ω. We can define a componentwise action of G on Ω × Ω. An orbital is an orbit of G on Ω × Ω. There is a unique diagonal orbital ∆ = {(α,α) : α ∈ Ω}, all others are called non-diagonal orbitals. For a non-diagonal orbital Γ we define the orbital graph Γ to be an undirected graph with vertex set Ω, and edge set the pairs in the orbital Γ. A famous theorem of Donald Higman states that the non-diagonal orbital graphs are all connected if and only if the action of G is primitive. Hence we can define the orbital diameter of a primitive permutation group to be the supremum of the diameters of its orbital graphs.
In 2010, Martin Liebeck, Dugald MacPherson and Katrin Tent classified infinite families of primitive permutation groups such that there is a t ∈ N which is an upper bound on the orbital diameter of all groups in the family. Their motivation and methods of proof were model theoretical and they provided no explicit bounds on the orbital diameter. Hence two natural goals in the study of the orbital diameters are to find explicit bounds and to classify groups with small orbital diameter.
In this thesis we provide important background information and detail the progress made towards finding explicit bounds on the orbital diameter. In particular, we consider the orbital diameter of the groups of simple diagonal type and their connection to the covering number of finite simple groups. Then we study the orbital diameter of affine groups, which provides a nice connection to the representation theory of quasisimple groups.