Let G be a cyclic group with generator a, and let G' be a group isomorphic to G. If phi: G-> G' is an isomorphism, show that for every x in G, phi(x) is completely determined by the value phi(a). That is, if phi: G -> G' and psi: G -> G' are two isomorphisms such that phi(a) = psi(a) and phi(x) = psi(x) for all x in G.
I know that it has something to do with how x = a^n for some integer. But I have no idea what to do next! Thanks in advance.