ABSTRACT
Each map can be associated with a unique element in a particular group and a finite set, where the action of the group on the set satisfies certain conditions. This association is given as an axiomatization, one for maps in orientable surfaces and another for maps in locally orientable surfaces, and in each case a permutation representation is given. The listing of maps is then reduced to the listing of permissible permutations. By specifying a particular action of these permutations on a given set that satisfies the conditions and, by adopting certain conventions, these permutations can be put in one-to-one correspondence with maps with labels attached to edges. Such maps are called labelled maps. A rooted map is recovered from a labelled map by erasing the attached labels and assigning a root in a canonical way.
