I am currently working on a problem trying to prove that there is no subgroup $ H $ of $\mathbb{Q}$ such that $H \cong \mathbb{Z} \times \mathbb{Z}$.
I was able to show that $\mathbb{Q}$ cannot be isomorphic to $ \mathbb{Z} \times \mathbb{Z}$, since $ \mathbb{Z} \times \mathbb{Z} $ can be generated by two elements, where as $\mathbb{Q}$ cannot be finitely generated. Also the subgroup $\mathbb{Z}$ of $\mathbb{Q}$ can also not be isomorphic since it can be generated by a single element.
However, I am somewhat stuck showing, that there cannot exist any subgroup of $\mathbb{Q}$ isomorphic to $ \mathbb{Z} \times \mathbb{Z} $. Some help would be very much appreciated.
I am also wondering, if it is also true that $\mathbb{R}$ and $\mathbb{C}$ have no subgroup isomorphic to $ \mathbb{Z} \times \mathbb{Z} $. I would guess that they do not have one since $\mathbb{Q}$ has none.
Hint: Suppose that $f:\Bbb Z\times \Bbb Z \to H$ is an isomorphism. Take $\frac{a}{b}=f(1,0)$ and $\frac{c}{d}=f(0,1)$. Then $\langle \frac{a}{b},\frac{c}{d}\rangle = \{n \frac{a}{b}+m\frac{c}{d}:n,m\in\Bbb Z\}$ would be isomorphic to $\Bbb Z \times \Bbb Z$, but $\langle \frac{a}{b},\frac{c}{d}\rangle$ is cyclic.