Regular maps with an alternating or symmetric group as automorphism group

Abstract

This paper provides a complete determination of which of the alternating groups An and the symmetric groups Sn occur as the automorphism group of some regular or chiral map on an orientable surface, and which of them occur as the automorphism group of a regular map on a non-orientable surface. The situation for some given types (m,k) is also considered, where k is the valency and m is the face-size, with special focus on types with m=3, and more particularly with (m,k)=(3,7) or (3,8), or their duals. Some observations are made also about what happens for regular and orientably-regular maps with given valency, and for regular and chiral polyhedra. Much but certainly not all of what is presented follows from theorems in previous papers by the author and others, and this one brings them and some new observations together into a single reference.