Question

Prove that every element of $\mathrm{\mathbb{Q}}/\mathrm{\mathbb{Z}}$ has finite order.

Short answer

It is proved that every element of$\mathrm{\mathbb{Q}}/\mathrm{\mathbb{Z}}$has finite order.

Step-by-step solution

Definition of the additive group. 

Additive Group:

An additive group is a group of elements under the operation of addition.

It is an abelian group, and it is generally written using the symbol + for its binary operation.

Proving that every element of ℚ/ℤ has finite order 

As we know that $\mathrm{\mathbb{Z}}$is an additive group. Considering $\mathrm{q}=\frac{\mathrm{m}}{\mathrm{n}}$where $\mathrm{m},\mathrm{n}\in \mathrm{\mathbb{Z}}$, we get $\mathrm{q}+\mathrm{\mathbb{Z}}\in \mathrm{\mathbb{Q}}/\mathrm{\mathbb{Z}}$

Then, every element of $\mathrm{\mathbb{Q}}/\mathrm{\mathbb{Z}}$can be given as $\mathrm{q}+\mathrm{\mathbb{Z}}$.

So,$\mathrm{n}(\mathrm{q}+\mathrm{\mathbb{Z}})=\mathrm{m}+\mathrm{\mathbb{Z}}=\mathrm{\mathbb{Z}}$= identity of$\mathrm{\mathbb{Q}}/\mathrm{\mathbb{Z}}$

Therefore, we get that $|\mathrm{q}+\mathrm{\mathbb{Z}}|\le 1$

This shows that$\mathrm{q}+\mathrm{\mathbb{Z}}$ has finite order, hence it is proved that the element of$\mathrm{\mathbb{Q}}/\mathrm{\mathbb{Z}}$ has finite order.