Automorphisms of imbedded graphs

If a linear graph is imbedded in a surface to form a map, then the map has a group of automorphisms which is a subgroup (usually, a proper subgroup) of the automorphism group of the graph. In this note it will be shown that, for any imbedding of K_n in an orientable surface, the order of the automorphism group of the resulting map is a divisor of n(n − 1), and that the order equals n(n − 1) if and only if n is a prime power. The explicit construction of imbeddings of K_Q, q = p^m with map automorphism group of order q(q − 1) gives rise to new types of regular map. There are also tenous connections with the theory of Frobenius groups.