Permutation groups with a cyclic regular subgroup and arc transitive circulants

Abstract

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 Γ of order n is one of the following: a complete graph K n , a lexicographic product Σ [K̄b]. A deleted lexicographic product Σ [K̄b] - bΣ, where ∑ is a smaller arc transitive circulant, or Γ is a normal circulant, that is, AutaΓ 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. © 2005 Springer Science + Business Media, Inc.