# Thread: Finding all the homomorphisms

1. ## Finding all the homomorphisms

Find all the homomorphisms from Z -> Z x Z and Z x Z -> Z.

For the first, I've found and proven the following.

$\phi$Z -> Z x Z such that a -> (a,0) or (0,a) or (0,0).

For the second one:

$\phi$Z x Z -> Z such that (a,b)-> a or b (the "forgetful" function) or 0.

So I have found three. Am I on the right track? I have verified that these are indeed a homomorphism.

2. There are still others, for example, $\phi(a) = (2a, 3a)$.

Instead of just trying to come up with examples of homomorphisms, begin with the assumption that $\phi: \mathbb{Z} \rightarrow \mathbb{Z} \times \mathbb{Z}$ is a homomorphism and see what you can deduce from that assumption alone.