# Reference request: a locally cyclic group is isomorphic to a section of the rational numbers

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. However, despite some searching, I have been unable to find a proof in the published literature on abelian groups.

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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Many thanks for these references. For torsion-free groups the result is mentioned on page 109 of Fuchs "Infinite Abelian Groups", Volume II. I cannot find the torsion case in Fuchs in any explicit form. Marshall Hall's "Theory of Groups" page 340 has "The additive group of rationals $R_+$ is a locally cyclic group which is aperiodic, and the group $R_+$ modulo $1$ is a periodic locally cyclic group. It is not too difficult to show that a locally cyclic group is a subgroup of one of these two groups." (No proof is given.) –  Mark Wildon Oct 4 '11 at 17:20