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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?

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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.

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  • $\begingroup$ 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) $\endgroup$ Commented Aug 1, 2022 at 17:39
  • $\begingroup$ 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. $\endgroup$ Commented Aug 1, 2022 at 17:40
  • $\begingroup$ @Dave, thanks for bringing this to my attention. I made a small addition to my reply. $\endgroup$
    – kabenyuk
    Commented Aug 2, 2022 at 4:22

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