Title:

Orbital diameters of primitive permutation groups

Let G be a transitive permutation group acting on a finite set X. Recall that G is primitive if there are no nontrivial equivalence relations on X which are preserved by G. An orbital graph of G is a graph with vertex set X and edges {x,y}, where (x,y) belongs to a fixed orbit of the natural action of G on the set X x X. A wellknown result by D.G. Higman asserts that G is primitive if and only if all the orbital graphs are connected. For a primitive group G, we define the orbital diameter of G to be the maximum of the diameters of all orbital graphs of G. Let C be an infinite class of finite primitive permutation groups. This gives rise to an infinite family of orbital graphs. It may be that the diameters of these orbital graphs tend to infinity. More interestingly, it may be that the diameters of all the orbital graphs are bounded above by some fixed constant; if this is the case, then we say that C is bounded. Previous results by M. W. Liebeck, D. Macpherson and K. Tent focus attention on classes of almost simple primitive permutation groups which are bounded. In the thesis we analyse the orbital diameters of three families of groups, as follows. Firstly, we analyse the alternating and symmetric groups. For the primitive actions of these groups, we give necessary numerical conditions for the orbital diameter to be bounded above by some constant c and we make the result precise for c=5. For each primitive action, we also describe either all or an infinite family of orbital graphs of diameter 2. Then we analyse the almost simple groups with socle isomorphic to the projective special linear group PSL(2,q). For the primitive actions of these groups we give necessary conditions for the orbital diameter to be bounded above by 2 and we also give information about orbital graphs of diameter 2. Lastly, we analyse a large family of simple groups known as the simple groups of Lie type which consist of the simple classical groups and exceptional groups. In particular, we analyse a class of primitive actions known as parabolic actions, giving a precise description of the actions for which the orbital diameter is at most 2. For the simple classical groups, we also describe infinite families of orbital graphs of diameter 2.
