On Partitioning the Orbitals of a Transitive Permutation Group

Let G be a permutation group on a set Omega with a transitive normal subgroup M. Then G acts on the set Orbl(M, Omega) of nontrivial M-orbitals in the natural way, and here we are interested in the case where Orbl( M,) has a partition P such that G acts transitively on P. The problem of characterising such tuples (M, G, Omega, P), called TODs, arises naturally in permutation group theory, and also occurs in number theory and combinatorics. The case where | P| is a prime-power is important in algebraic number theory in the study of arithmetically exceptional rational polynomials. The case where | P| = 2 exactly corresponds to self-complementary vertex-transitive graphs, while the general case corresponds to a type of isomorphic factorisation of complete graphs, called a homogeneous factorisation. Characterising homogeneous factorisations is an important problem in graph theory with applications to Ramsey theory. This paper develops a framework for the study of TODs, establishes some numerical relations between the parameters involved in TODs, gives some reduction results with respect to the G-actions on Omega and on P, and gives some construction methods for TODs.