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 ) ⊴ M o n ( 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 | M o n ( H ) | / d = k . This k will be called the duality coindex of the hypermap.