Regular embeddings of Kn,nKn,n where nn is an odd prime power

We show that if n=pen=pe where pp is an odd prime and e≥1e≥1, then the complete bipartite graph Kn,nKn,n has pe−1pe−1 regular embeddings in orientable surfaces. These maps, which are Cayley maps for cyclic and dihedral groups, have type {2n,n}{2n,n} and genus (n−1)(n−2)/2(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 nn. We deduce that if nn is odd then Kn,nKn,n has at least n/∏p|npn/∏p|np orientable regular embeddings, and that this lower bound is attained if and only if no two primes pp and qq dividing nn satisfy p≡1mod(q)p≡1mod(q).