On partitioning the orbitals of a transitive permutation group

Abstract:

Let $G$ be a permutation group on a set $\Omega$ with a transitive normal subgroup $M$. Then $G$ acts on the set $\mathrm {Orbl}(M,\Omega )$ of nontrivial $M$-orbitals in the natural way, and here we are interested in the case where $\mathrm {Orbl}(M,\Omega )$ has a partition $\mathcal P$ such that $G$ acts transitively on $\mathcal P$. The problem of characterising such tuples $(M,G,\Omega ,\mathcal P)$, called TODs, arises naturally in permutation group theory, and also occurs in number theory and combinatorics. The case where $|\mathcal P|$ is a prime-power is important in algebraic number theory in the study of arithmetically exceptional rational polynomials. The case where $|\mathcal P|=2$ 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 $\mathcal P$, and gives some construction methods for TODs.