The duality index of oriented regular hypermaps

By adapting the notion of a chirality group, the duality group of H can be defined as the minimal subgroup D(H)⊴Mon(H) such that H/D(H) is a self-dual hypermap (a hypermap isomorphic to its dual). Here, we prove that for any positive integer d, we can find a hypermap of that duality index (the order of D(H)), even when some restrictions apply, and also that, for any positive integer k, we can find a non-self-dual hypermap such that |Mon(H)|/d=k. This k will be called the duality coindex of the hypermap.