Show that $({\mathbb{Q}},+)$ is not finitely generated using the Fundamental Theorem of Finitely Generated Abelian Groups.

Question

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?

Answer 1

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.

Answer 2

If $\mathbb Q$ wants to be finitely generated, then it can't be divisible group. But it is.

Answer 3

Assume that $(\mathbb Q,+)$ is finitely generated. Then, by the fundamental theorem there exist $m,n\ge 0$ and $d_1\mid\cdots\mid d_m$ with $d_i>1$ such that $\mathbb Q\simeq \mathbb Z/d_1\mathbb Z\oplus\cdots\oplus\mathbb Z/d_m\mathbb Z\oplus \mathbb Z^n$. If $m\ge 1$, then there exists $x\in\mathbb Q$, $x\neq 0$, such that $d_1x=0$, a contradiction. Thus we get $m=0$. Then $\mathbb Q\simeq \mathbb Z^n$. If $n\ge 2,$ then there exist $x_1,x_2\in\mathbb Q$ which are linearly independent over $\mathbb Z$. But $x_1=a_1/b_1$ and $x_2=a_2/b_2$ give $(b_1a_2)x_1+(-b_2a_1)x_2=0$, a contradiction. So we must have $n=1$, that is, $\mathbb Q$ is cyclic. Assume that it is generated by $a/b$ with $b\ge 1$. Then $\frac{1}{b+1}$ can not be written as $\frac{ka}{b}$, and again we reached to a contradiction.

(Of course, there are simpler and much more natural arguments to show that $(\mathbb Q,+)$ is not finitely generated.)

Edit. In particular, this shows that the additive group of any field of characteristic $0$ is not finitely generated. (However the property holds for any infinite field.)

Answer 4

In a finitely generated $0$ characteristic Abelian group you cannot divide arbitrary many times by $2$, say.