On finite permutation groups with a transitive cyclic subgroup. (Q420674)

The authors study the structure of finite permutation groups with a transitive cyclic subgroup. We call such permutation groups c-groups. Finite primitive c-groups were classified by \textit{G. A. Jones} [in J. Group Theory 5, No. 4, 403-407 (2002; Zbl 1012.20002)], completing the work of Burnside, Galois, Ritt, Schur and Feit.NEWLINENEWLINE In this paper, the authors extend this classification to permutation groups that are (i) quasiprimitive, (ii) almost simple, and (iii) innately transitive. They also analyse the actions of normal subgroups of arbitrary groups with transitive cyclic subgroups. Finally, they give an application to circulant homogeneous factorisations of complete graphs. In this way, they obtain a new proof of their possible parameters together with some new structural information.