How to make sure if the set $\{z\in\mathbb{C}:\exists n\in\mathbb{N}\; z^n=1\}$ is a subgroup of $(\mathbb{C}\setminus\{0\},\cdot)$?

Question

How can I make sure if the set $\{z\in\mathbb{C}:\exists n\in\mathbb{N}\; z^n=1\}$ is a subgroup of $(\mathbb{C}\setminus\{0\},\cdot)$?

Answer 1

Call your set $S$. First of all $z \in S$ implies $z \neq 0$, so $S \subseteq \mathbb C \setminus\{0\}$. To prove $S$ is a subgroup, it is enough to show that $xy^{-1} \in S$ whenever $x,y \in S$.

$x \in S$, so there exists $n \in \mathbb N$ such that $z^n =1$. Similarly, there exists $m \in \mathbb N$ such that $y^m = 1$. Now take $r = mn$. We have $(xy^{-1})^r = (x^n)^m (y^m)^{-n} = 1$, and so $xy^{-1} \in S$.