Prove that $\mathbb{Q} {/ \mathbb{Z}}$ contains elements of every possible finite order.

Question

I was able to prove that every element of $\mathbb{Q} {/ \mathbb{Z}}$ has finite order. Now I'm not sure how to prove this one. I usually start my proofs by writing down an equation or end statement of what I have to prove. Any hints on where to start?

Answer 1

The element $\frac{1}{n}$ has order $n$

Answer 2

Using the first isomorphism theorem, this quotient ring $\mathbb{Q} {/ \mathbb{Z}}$ is isomorphic to the multiplicative group $\{z\in \mathbb{z}:z^n=1, for~ some~ n \in \mathbb{N}\}$, this group contains all the primitive $n-th$ roots of unity for any $n$ which are of the order $n$.

from direct computation, as Jorge mentioned, the coset $\frac{1}{n}+\mathbb{Q}$ will be of the order $n$