# How to find a finite presentation of a finitely generated abelian group?

Yesterday I asked a question which turned out to be pretty bad posed due to the fact that I am not really experienced with presentations of groups. However, I was told the following result:

If an abelian group $$G$$ is finitely generated, then $$G$$ is finitely presented.

I thought that I know how to prove this, but I keep struggling with it and I am not able to do it. I considered $$G$$ to be generated by the set $$X=\{x_1, x_2, ... , x_n\}$$. Then I took the free abelian group on $$X$$ and I considered the map that sends $$X$$'s generators to those of the free abelian group. This gave me a homomorphism between $$G$$ and the free abelian group on $$X$$. However, I don't know what to do next to get that $$G$$ has a finite presentation. I also don't know if my approach is fruitful. Sorry if this is well-known, but I could find no references.

Let $$F$$ be the free abelian group generated by symbols $$x_1,\ldots,x_n$$, and let $$\phi:F \to G$$ be the homomorphism that takes each $$x_i$$ to the corresponding generator of $$G$$. Then the kernel $$K$$ of $$\phi$$ is generated by a finite subset $$R$$ of $$F$$.
Then a presentation of $$G$$ is $$\langle x_1,\ldots,x_n \mid R \cup C \rangle,$$ where $$C$$ consists of relations $$x_ix_j=x_jx_i$$ for all $$1 \le i,j \le n$$. The relations in $$C$$ ensure that $$G$$ is abelian).