On finite permutation groups with a transitive cyclic subgroup

In this paper, we study the structure of finite permutation groups with a transitive cyclic subgroup. In particular we extend the classification due to W. Feit and G.A. Jones of the primitive groups containing such subgroups to permutation groups that are (i) quasiprimitive, (ii) almost simple, and (iii) innately transitive. We also analyse the actions of normal subgroups of arbitrary groups with transitive cyclic subgroups. We give an application to circulant homogeneous factorisations of complete graphs, obtaining new structural information and giving a new proof of their possible parameters.