$G$ is a cyclic group of order $n$ and there is an integer $m$ that divides $n$.Prove that there is a subgroup of $G$ of order $\frac{n}{m}$.
Let $G=\langle a\rangle$ be a cyclic group of order $n$. Show that, for every divisor $d$ of $n$, there exists a subgroup of $G$ whose order is $d$.