Regular embeddings of where is an odd prime power

We show that if n = p e where p is an odd prime and e ≥ 1 , then the complete bipartite graph K n , n has p e − 1 regular embeddings in orientable surfaces. These maps, which are Cayley maps for cyclic and dihedral groups, have type { 2 n , n } and genus ( n − 1 ) ( n − 2 ) / 2 ; one is reflexible, and the rest are chiral. The method involves groups which factorise as a product of two cyclic groups of order n . We deduce that if n is odd then K n , n has at least n / ∏ p | n p orientable regular embeddings, and that this lower bound is attained if and only if no two primes p and q dividing n satisfy p ≡ 1 mod ( q ) .