Consider $\displaystyle (\mathbb Q,+)$ and $\displaystyle (\mathbb Q/\mathbb Z,+).$ Let $\displaystyle n\ge1$ and define $\displaystyle G_n=\left( \left[ \dfrac{a}{n} \right],a\in \mathbb{Z} \right).$ Prove that $\displaystyle G_n$ is a subgroup of $\displaystyle \mathbb Q/\mathbb Z,$ and that it has order $\displaystyle n.$

$\displaystyle \left[\dfrac an \right]$ is the class of $\displaystyle \dfrac an.$******