Permutation Groups with a Cyclic Regular Subgroup and Arc Transitive Circulants

A description is given of finite permutation groups containing a cyclic regular subgroup. It is then applied to derive a classification of arc transitive circulants, completing the work dating from 1970’s. It is shown that a connected arc transitive circulant of order n is one of the following: a complete graph Kn , a lexicographic product [K̄b], a deleted lexicographic product [K̄b] − b , where is a smaller arc transitive circulant, or is a normal circulant, that is, Aut has a normal cyclic regular subgroup. The description of this class of permutation groups is also used to describe the class of rotary Cayley maps in subsequent work.