Regular orientable imbeddings of complete graphs

This paper classifies the regular imbeddings of the complete graphs K_n in orientable surfaces. Biggs showed that these exist if and only if n is a prime power p^e, his examples being Cayley maps based on the finite field F = GF(n). We show that these are the only examples, and that there are $\text{φ(n ? 1)}\text{e}$ isomorphism classes of such maps (where φ is Euler's function), each corresponding to a conjugacy class of primitive elements of F, or equivalently to an irreducible factor of the cyclotomic polynomial Φ_{n ? 1}(z) over GF(p). We show that these maps are all equivalent under Wilson's map-operations H_i, and we determined for which n they are reflexible or self-dual.