# Is this statement true for divisible Groups?

If $G$ and $H$ are divisible groups each of which is isomorphic to a subgroup of the other, then $G$ is isomorphic to $H$.

Here, $G$ and $H$ are abelian groups. Can we assume another adjective rather than divisibility?

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1) Cyclic. 2) Finitely-generated-and-torsion-free. Surely many others (perhaps "finitely generated" suffices?) –  Alon Amit May 8 '11 at 13:47