Let $(\mathbb Q\setminus\{0\},*)$ be a group of the rational numbers under multiplication and $(\mathbb Q,+)$ be the group of rational numbers.

Question

Let $(\mathbb Q\setminus\{0\},*)$ be a group of the rational numbers under multiplication and $(\mathbb Q,+)$ be the group of rational numbers under addition. Find the order of every element of this group.

Can somebody help me with this question? I don't know where to start. I would really appreciate some help

Answer 1

Which $n\in\mathbb N\setminus\{0\}$ and $q\in \mathbb Q\setminus\{0\}$ solve $q^n=1$?

Which $n\in\mathbb N\setminus\{0\}$ and $q\in \mathbb Q$ solve $nq=0$?