Here it is:

I have been able to prove such a homomorphism exists with a cyclic subgroup of H anyway. How do I extend this to they whole group H? For example when I do this using $\displaystyle H = D_8$ I end up with a "homomorphism" that is a relation, not a function.Show that if H is any group and h is an element of H with $\displaystyle h^n = e$, then there is a unique homomorphism from $\displaystyle Z_n = < x >$ to H such that $\displaystyle x \mapsto h$.

And as far as uniqueness goes the homomorphism maps x to h. Isn't that the only function that will do that? I mean if we have $\displaystyle \phi : Z_n \to H: x \mapsto h$ how could we have a distinct function $\displaystyle \psi : Z_n \to H: x \mapsto h$? If they both map x to h wouldn't they have to be the same function automatically? I'm missing something here...

-Dan