Example of subgroup of $\mathbb Q$ which is not finitely generated

Question

I was looking for a proper subgroup of $\mathbb Q$ which is not finitely generated under the addition operation.

We know every finitely generated subgroup of $\mathbb Q$ is cyclic. For a proper subgroup I am just thinking about the subgroup $H$ generated by $\{\frac{1}{p} : p \text{ prime }\}$ may work. It seems $1/4$ is not in $H.$ Is this a correct example? Thanks

Answer 1

Try $\mathbb Z[\frac12]$, that is, the set of binary fractions.

More generally, $\mathbb Z[\frac1p]$, that is, the set of fractions whose denominators are powers of the prime $p$.