I must prove that $\frac{\Bbb{Z} \times \Bbb{Z}}{\langle(3,3)\rangle}$ is isomorphic to $\Bbb{Z} \times \Bbb{Z_3}$.
I am trying to do this using the first isomorphism theorem ie. for $\phi: G \to H$ a homomorphism, we have that:
$$\frac{G}{\operatorname{Ker}(\phi)} \cong \operatorname{Im}(\phi) \subset H$$
However, I just can't seem to be able to find the right homomorphism that will give me the kernel of $\phi$ that I need. If someone can point out what the homomorphism should be, it would help greatly.
Thanks!
Just in case these differ in any way, here are some of the definitions we have:
$\langle(3,3)\rangle$ is the generating set of $(3,3)$
$\Bbb{Z}_3$ is the integers modulo $3$
Product of groups, kernel and image defined as 'normal'.