A group $G$ is *locally cyclic* if whenever $H \le G$ is a finitely generated subgroup then $H$ is cyclic. If $G$ is a locally cyclic group then $G$ is isomorphic to a quotient of a subgroup of the rational numbers under addition. An online proof of this fact appears at [groupprops](http://groupprops.subwiki.org/wiki/Locally_cyclic_iff_subquotient_of_rationals). However, despite some searching, I have been unable to find a proof in the published literature on abelian groups.



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> Is there a published paper or textbook that has a proof that every locally cyclic group is isomorphic to a quotient of a subgroup of the rational numbers?
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Reference request: a locally cyclic group is isomorphic to a section of the rational numbers
For torsion-free groups it is proved in Kurosh, "Group theory", See 3d edition, Chapter VIII, Section 30 (of course the result can be found in the 1st edition as well). Oroginally it was proved in Reinhold Baer, "Abelian groups without elements of finite order". Duke Math J. 3 (1): 68–122, 1937. For groups with torsion, it follows from old results as well but I am not sure anybody specifically mentioned it somewhere. Of course you need to look at Fuchs, "Infinite Abelian groups" (both volumes). If it is not there, it is probably not anywhere else.


