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 Γ of order n is one of the following: a complete graph Kn, a lexicographic product \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Sigma [{\bar K}_b]$\end{document}, a deleted lexicographic product \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Sigma [{\bar K}_b] - b\Sigma$\end{document}, where Σ 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.