Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.21
Prove that $\mathbb{Q}$ has no proper subgroups of finite index. Deduce that $\mathbb{Q}/\mathbb{Z}$ has no proper subgroups of finite index. [Recall Exercise 1.6.21 and Exercise 3.1.15.]
Solution: We begin with a lemma.
Lemma: If $D$ is a divisible abelian group, then no proper subgroup of $D$ has finite index.
Proof: We saw previously that no finite group is divisible and that every proper quotient $D/A$ of a divisible group is divisible; thus no proper quotient of a divisible group is finite. Equivalently, $[D:A]$ is not finite. $\square$
Because $\mathbb{Q}$ and $\mathbb{Q}/\mathbb{Z}$ are divisible, the conclusion follows.
