Problem: Let $p$ be a positive prime integer. Consider the set under multiplication $$G := \{x \in \mathbb{C} : ∃n \in \mathbb{N}, x^{p^n} = 1\}.$$ Show that every proper subgroup of $G$ is finite.
I know that $G$ must be infinite or else, assume that $|G|=m$. We can choose $n \in \mathbb{N}$ such that $p^n>m$. The polynomial $x^{p^n}-1$ has precisely $p^n$ distinct roots which belong to $G$. Thus, $m=|G|\geq p^n$, leads to a contradiction. However, I have no idea to prove that every proper subgroup of $G$ must be finite.
Help me, thank you so much!