TITLE: Show that $({\mathbb{Q}},+)$ is not finitely generated using the Fundamental Theorem of Finitely Generated Abelian Groups.
QUESTION [20 upvotes]: Can anyone please help me out on how to use the fundamental theorem of finitely generated abelian groups to prove that $({\mathbb{Q}},+)$ is not finitely generated?

REPLY [16 votes]: As per the OP's request, here's an explanation of my comment:

I don't see why you need that theorem (FToFGAG). It couldn't be finitely generated since, letting p be any prime greater than the product of the denominators of the generators in lowest form, one could not generate 1/p. 

Suppose $(\mathbb{Q}, +)$ were finitely generated.  Let
$$\left\{\frac{n_1}{d_1}, \dots, \frac{n_k}{d_k}\right\}$$
be a generating set.  Let $p$ be any prime that doesn't divide $\prod_{i=1}^k d_i$.  Then clearly one cannot generate $\frac 1p$ by adding and subtracting (whole number multiples of) the elements of the generating set, contradicting the fact that it's supposed to be a generating set.
To be explicit, let's consider an arbitrary ($\mathbb{Z}$-linear) combination of the elements in the generating set:
$$m_1\frac{n_1}{d_1}+ \cdots + m_k\frac{n_k}{d_k}=\frac{\mathrm{long\ expression}}{\prod_i^kd_i}$$
There's no way to reduce such a fraction to $\frac 1p$ if $p$ doesn't divide the denominator.