How do we conclude that following two groups are isomorphic? Let $H= \Bbb Z  \times \Bbb Z \times \{0\}$ and $T = \{0\} \times \Bbb Z \times \{0\}$
Then $H/T $ is isomorphic to $\Bbb Z$
$\Bbb Z$ denotes integers, and $\times$ Cartesian product.
How did we make that conclusion? I come across such examples very often, could you give me some tips in order to see such I believe "triviality"?

Define a map $\phi:H \to \mathbb{Z}$ such that $(m,n,0) \mapsto m$. It is easily verified that this is a group homomorphism, and that the image of $\phi$ is all of $\mathbb{Z}$. Moreover note that $\ker{\phi} = T$. Therefore, by the First Isomorphism theorem we have that:
$$
H/T \cong \text{Im}{\phi} = \mathbb{Z}
$$
