Problem 4
Consider the additive quotient group $\mathbb{Q}/\mathbb{Z}$.
1. Show that every coset of $\mathbb{Z}$ in $\mathbb{Q}$ contains exactly one representative $q \in \mathbb{Q}$ in the range
$0 \le q < 1$.
2. Show that every element of $\mathbb{Q}/\mathbb{Z}$ has finite order but that there are elements of arbitrarily
large order.
3. By the previous problem, $\mathbb{Q}/\mathbb{Z}$ is a subgroup in $\mathbb{R}/\mathbb{Z}$. Prove that $\mathbb{Q}/\mathbb{Z} = \{x \in \mathbb{R}/\mathbb{Z}, \text{s.t. } |x| \text{ is finite}\}$.
