Regular p-Maps and Regular Cayley Maps of Znp

Abstract

A (topological) map is a cellular decomposition of a closed surface. A common way to describe maps is to view them as 2-cell embeddings of graphs. A map is called a p-map if it has a prime p-power vertices. An orientably -regular (resp. A regular) p -map is called solvable if the group G+ of all orientation-preserving automorphisms (resp. the group G of automorphisms) is solvable; and called normal if G+ (resp. G) contains the normal Sylow p-subgroup. In this talk, it will be proved that both orientably-regular p-maps and regular p-maps are solvable and except for few cases that p ∈ 2, 3, they are normal.Regular Cayley Maps are regular embeddings of Cayley graphs. One of a special families of regular p-maps is regular Cayley maps of elementary abelian p-groups. In this talk, a complete classification of such maps will be given and moreover, the number of these maps and their genera will be enumerated.