$\mathbb{Z} \times \mathbb{Z}$ and application of isomorphism theorems Let $G = \mathbb{Z} \times \mathbb{Z}$ with group law given by addition. Let $H$ be the subgroup generated by the element $(2,3)$. Then $G/H$ is isomorphic to $\mathbb{Z}$. 
Is $G/H$ also isomorphic to $\mathbb{Z}_2\times \mathbb{Z}_3$
using the homomorphism $h((x,y)) = (x\mod 2,\ y\mod 3)$? (here, $h((0,0)) = (0,0)$ and $h((x,y) + (m,n)) = h((x,y)) + h((m,n))$.

You have that $(2,3)H$ and $(4,3)H$ are both send to the same element using the homomorphism. However it's not true that $(2,3)H = (4,3)H$, as this would mean that $(2,0) \in H$, which isn't the case.
Hence $G/H$ isn't isomorphic to $\mathbb{Z}_2 \times \mathbb{Z}_3$.
