Help with proof on finitely generated abelian groups and its product of cyclic groups

Question

In my lecture notes, there is a corollary stating that if $G$ is a finitely generated abelian group, then there is a surjective homomorphism $f: \mathbb{Z} ^n\rightarrow G$ for some integer $n$. It then uses this to state:

We need to prove that if $H \cong \mathbb{Z} ^m$ is a subgroup of $ \mathbb{Z} ^n$, then $ \mathbb{Z} ^n/H$ is isomorphic to a product of finitely many cyclic groups.

So I have two questions really;

1. How can we even take this quotient to begin with? Wouldn't it correspond to the addition/multiplication of tuples of different sizes?
2. How do we know every finitely generated abelian group can be written in the form of this quotient?

Answer 1

1. $H$ is a subgroup of $\mathbb Z^n$ that is isomorphic to $\mathbb Z^m$. This does not mean that $H$ equals $\mathbb Z^m$. For example, you could have $$H = \mathbb Z^m\times \underbrace{\{0\} \times \{0\} \times \cdots \times \{0\}}_{n - m\text{ times}}.$$
2. Use the first isomorphism theorem. You already know that there is a surjective homomorphism $f : \mathbb Z^n \to G$. If we let $H = \ker f$, then $G \cong \mathbb Z^n / H.$