What are all the finite subgroups of $(\mathbb{R} \setminus \{0\},\cdot)$

Question

I tried attempting the question but I really didn't know where to start. Any help regarding this question would be great.

Answer 1

The most helpful subgroup criterion to think about here is closure. If $S \subseteq \mathbb{R} - \{0\}$, for $S$ to be a finite subgroup implies that for all $x \in S$, there exists $n \in \mathbb{N}$ with $n \neq 1$ such that $x = x^n$, otherwise the powers of x form an infinite subset of S (since S is a subgroup and hence closed), which implies that S is not finite.

So the question boils down to: for which $x \in \mathbb{R} - \{0\}$ do we have $x = x^n$ for some $n \in \mathbb{N}$ with $n \neq$ 1?

Answer:

This only holds for 1 and -1, as for all other $x \in \mathbb{R} - \{0\}$, $|x| \neq |x|^n$, so $x \neq x^n$. It's easy to check the subgroup criteria for the groups generated by the powers of 1 and the powers of -1.