Classification of the regular oriented hypermaps with a prime number of hyperfaces

Abstract

Regular oriented hypermaps are triples (G; a; b) consisting of a nite 2-generated group G and a pair a, b of generators of G, where the left cosets of ⟨a⟩, ⟨b⟩ and ⟨ab⟩ describe respectively the hyperfaces, hypervertices and hyperedges. They generalise regular oriented maps (triples with ab of order 2) and describe cellular embeddings of regular hypergraphs on orientable surfaces. In [5] we have classi ed the regular oriented hypermaps with prime number hyperfaces and with no non-trivial regular proper quotients with the same number of hyperfaces (i.e. primer hypermaps with prime number of hyperfaces), which generalises the classi cation of regular oriented maps with prime number of faces and underlying simple graph [13]. Now we classify the regular oriented hypermaps with a prime number of hyperfaces. As a result of this classi cation, we conclude that the regular oriented hypermaps with prime p hyperfaces have metacyclic automorphism groups and the chiral ones have cyclic chirality groups; of these the \canonical metacyclic" (i.e. those for which ⟨a⟩ is normal in G) have chirality index a divisor of n (the hyperface valency) and the non \canonical metacyclic" have chirality index p. We end the paper by counting, for each positive integer n and each prime p, the number of regular oriented hypermaps with p hyperfaces of valency n.