Thomas observed the following generalization of Problem 1 on HW #4: if H is a cyclic subgroup of the symmetric group S_n, with |H| = n, then the quotient N_Sn(H)/C_Sn(H) is isomorphic to Aut(H). (In general, the N/C theorem implies N_G(H)/C_G(H) is isomorphic to a subgroup of Aut(H).Recall, if H is cyclic, then Aut(H) is isomorphic to the group of units U(Z_n) of Z_n, of order phi(n) (the Euler totient function). By considering a fixed generator sigma of H, and its powers, one sees that all generators of H have the same cycle-type as sigma, and hence are all conjugate to sigma in S_n. The claim follows immediately from this observation.