On Orbital Partitions and Exceptionality of Primitive Permutation Groups

Let G and X be transitive permutation groups on a set Ω such that G is a normal subgroup of X. The overgroup X induces a natural action on the set Orbl(G,Ω) of non-trivial orbitals of G on Ω. In the study of Galois groups of exceptional covers of curves, one is led to characterizing the triples (G,X,Ω) where X fixes no elements of Orbl(G,Ω); such triples are called exceptional. In the study of homogeneous factorizations of complete graphs, one is led to characterizing quadruples (G,X,Ω,P) where P is a partition of Orbl(G,Ω) such that X is transitive on P; such a quadruple is called a TOD (transitive orbital decomposition). It follows easily that the triple (G,X,Ω) in a TOD (G,X,Ω,P) is exceptional; conversely if an exceptional triple (G,X,Ω) is such that X/G is cyclic of prime-power order, then there exists a partition P of Orbl(G,Ω) such that (G,X,Ω,P) is a TOD. This paper characterizes TODs (G,X,Ω,P) such that XΩ is primitive and X/G is cyclic of primepower order. An application is given to the classification of self-complementary vertex-transitive graphs.