Combinatorial formulas, invariants and structures associated with primitive permutation representations of PSL(2,q) and PGL(2,q)

This thesis describes various aspects of primitive permutation representations of the groups PSL(2,q) and PGL(2,q). The disjoint cycle structures, ranks, cycle index formulas and the subdegrees of these representations are computed. A method is devised for constructing some suborbital graphs of PSL(2,q) and PGL(2,q) on the cosets of their maximal dihedral subgroups of orders q-1 and 2(q-l) respectively. Some graph theoretic properties such as the girth and diameter are discussed for some of these graphs. A general form of the intersection matrix of PGL(2,q) on the cosets of its maximal dihedral subgroup of order 2(q-1) relative to the suborbit of length 2(q-1) is given. The number of triangles on every edge of the suborbital graph corresponding to this intersection matrix is shown to be q-1. - iv -