Regular Cayley maps for finite abelian groups

Authors:
Marston Conder;Robert Jajcay;Thomas Tucker
Affiliations:
Department of Mathematics, University of Auckland, Auckland, New Zealand 92019;Department of Mathematics and Computer Science, Indiana State University, Terre Haute, USA 47809;Mathematics Department, Colgate University, Hamilton, USA 13346
Venue:
Journal of Algebraic Combinatorics: An International Journal
Year:
2007

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Abstract

A regular Cayley map for a finite group A is an orientable map whose orientation-preserving automorphism group G acts regularly on the directed edge set and has a subgroup isomorphic to A that acts regularly on the vertex set. This paper considers the problem of determining which abelian groups have regular Cayley maps. The analysis is purely algebraic, involving the structure of the canonical form for A. The case when A is normal in G involves the relationship between the rank of A and the exponent of the automorphism group of A, and the general case uses Ito's theorem to analyze the factorization G = AY, where Y is the (cyclic) stabilizer of a vertex.