Regular Cayley maps for finite abelian groups

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.

Finite edge-transitive Cayley graphs and rotary Cayley maps

“…This fact was also observed in [8] This proposition is proved in Subsection 3.3. Taking normal quotient of Cayley graphs also provides a characterisation of rotary Cayley maps of abelian groups; see Proposition 5.5.…”

“…, and by Corollary 6.6, either C is a Singer subgroup of X, or X = PΓL (2,8) and C ∼ = Z 9 . For the latter, there is no involution x ∈ X such that a x = a −1 ; see the Atlas [9].…”

“…A map M is called a Cayley map of a group G if AutM contains the regular subgroupĜ. The study of orientable Cayley maps was initiated by Biggs [1] in 1972, and is a current active topic in topological and algebraic graph theory; see for example [3,8,33,34]. Here we apply Theorem 1.5 to give a characterisation of rotary Cayley maps of simple groups.…”

“…Such Cayley maps have been sought for a long time; see [8,33]. We first construct an infinite family of regular Cayley maps.…”

“…The structure and embedding of rotary Cayley maps of abelian groups were investigated in a recent paper [8] in terms of the given base group. Proposition 5.5 tells us that for such a Cayley map, changing the base group may lead to a nicer description of the map than using the original base group.…”