Permutation Groups with a Cyclic Two-Orbits Subgroup and Monodromy Groups of Siegel Functions

We classify the finite primitive permutation groups which have a cyclic subgroup with two orbits. This extends classical topics in permutation group theory, and has arithmetic consequences. By a theorem of C. L. Siegel, affine algebraic curves with infinitely many integral points are parametrized by rational functions whose monodromy groups have this property. We classify the possibilities of these monodromy groups, and give an application to Hilbert's irreducibility theorem.