# Classification of subgroups of $(\mathbb Q,+)$ that are not finitely generated. [closed]

Proper subgroups of $$(\mathbb Q,+)$$ that are not finitely generated exist, e.g., Example of subgroup of $$\mathbb Q$$ which is not finitely generated.

Is there a classification of subgroups of $$(\mathbb Q,+)$$ that are not finitely generated?

Let $$P$$ be the set of all prime numbers and let $$f: P\to\mathbb{N}\cup\infty$$ be an arbitrary function. Let $$N_f=\{n\in\mathbb{N}\mid n=\prod_{p\in P} p^{k_p}, k_p\leq f(p)\}.$$ Let $$\mathbb{Q}_f=\{a/b\mid a\in\mathbb{Z},b\in N_f\}$$.

Theorem. For any function $$f$$ the set $$\mathbb{Q}_f$$ is a subgroup.

If $$H$$ is a subgroup of $$\mathbb{Q}$$, then there exists a function $$f$$ and an non-negative integer $$m$$ that $$H=m\mathbb{Q}_f$$.

The subgroup $$m\mathbb{Q}_f$$ is finitely generated (and hence cyclic) if and only if $$f(p)<\infty$$ for every prime $$p$$ and $$f(p)>0$$ only for a finite number of prime.

Examples.

1. If $$f(2)=\infty$$, $$f(p)=0$$ for all $$p\in P$$, $$p\neq2$$, then we obtain a subgroup $$\mathbb{Q}_2=\mathbb{Z}[1/2]$$ (see Example of subgroup of $$\mathbb{Q}$$ which is not finitely generated).

2. If $$f(p)=\infty$$ for each $$p\in P$$, then $$\mathbb{Q}_f=\mathbb{Q}$$.

3. If $$f(p)=1$$ for each $$p\in P$$, then $$\mathbb{Q}_f=\operatorname{gr}(1/p\mid p\in P)$$.

4. If $$f(2)=a>0$$ and $$f(p)=0$$ for each $$p\neq2$$, then $$\mathbb{Q}_f=\operatorname{gr}(1/2^a)$$ is a cyclic group.

You can read about it in How to find all subgroups of $$\mathbb{Q}$$ and in the article 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.

Adding. In this connection it seems useful to read the chapter 'Torsion-Free Groups' of Laszlo Fuchs' book Abelian Groups. Any edition (and there are many) of this book has such a chapter.

• FYI I glanced through my copy of Abelian Groups by Fuchs (1960), looking at Subject Index listings for "rational group" (pp. 25, 149, 171, 211, 270), and found a possible lead in a in a 1937 Duke Mathematical Journal paper -- Abelian groups without elements of finite order by Baer -- but only the first page (continued) Aug 1, 2022 at 17:39
• is viewable and it wasn't clear to me that Baer's paper was worth mentioning. However, if I had looked through Fuchs' Bibliography (I didn't have the time or interest), then I would have seen the Beaumont/Zuckerman paper near the bottom of the first page of the Bibliography, and its title would definitely led me to investigate further. Aug 1, 2022 at 17:40
• @Dave, thanks for bringing this to my attention. I made a small addition to my reply. Aug 2, 2022 at 4:22