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 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 , then there is a unique homomorphism from to H such that .

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 how could we have a distinct function ? If they both map x to h wouldn't they have to be the same function automatically? I'm missing something here...

-Dan