How to find all subgroups of $(\mathbb{Q},+)$

Question

Exercise 1.6.13 from Scott's book Group Theory.

(Hard) Find all subgroups of $(\mathbb{Q},+)$. Hint: It is slightly easier to find those subgroups $H$ such that $1\in H.$

I've found some of those subgroups: $\mathbb{Z}$, $\mathbb{Q}$, $\langle 1,\frac{1}{2}\rangle$, $\langle 1,\frac{1}{2},\frac{1}{3}\rangle$, $\cdots $. But can't I find all of them. How to find them?

Answer 1

Let $A$ be an additive subgroup of $\mathbb Q$. Let $D$ be the set of denominators occurring in $A$ when you consider reduced fractions only. Then the following are easy to prove:

• $b\in D, u \mid b \implies u\in D$
• $u,v \in D, (u,v)=1 \implies uv \in D$

The first property means that you cannot increase the power of a prime in $D$. The second property means that you can combine powers of different primes. That should give you a description of $D$.

$A \cap \mathbb Z$ is an additive subgroup of $\mathbb Z$ and these are easy to characterize. That should give you a description of the set $N$ of numerators in $A$.

For a complete solution, see the paper

Ross A. Beaumont and H. S. Zuckerman, A characterization of the subgroups of the additive rationals, Pacific J. Math. Volume 1, Number 2 (1951), 169-177. MR0044522