$\mathbb Q/\mathbb Z$ is an infinite group I'm trying to prove that $\mathbb Q/\mathbb Z$ is an infinite abelian group, the easiest part is to prove that this set is an abelian group, I'm trying so hard to prove that this group is infinite without success. 
This set is defined to be equivalences classes of $\mathbb Q$, where $x\sim y$ iff $x-y\in \mathbb Z$.  
I need help here.
thanks a lot

The map $\rho: \mathbb{R} \to S^1,\;t \mapsto e^{2\pi\text{i}t}$ is  a surjective homomorphism of groups with kernel $\mathbb{Z}$. Hence $\rho(\mathbb{Q})$ is the group of roots of unity $\mu$ and $\mathbb{Q}/\mathbb{Z}\cong \mathbb{\mu}$. In particular, its infinite, since $\mu$ has the infinite subset $\{e^{\frac{2\pi\text{i}}{n}}\mid n \in \mathbb{Z},\;n> 0\}$. 
