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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

1 Answer 1

These are called Cantor-Bernstein theorems. The result for divisible groups was extended to injective modules in Bumby (1965). A study of these theorems is made in Wisbauer (2004), and in particular Artinian modules and cohopfian uniserial modules have this property. If G,H are both assumed to have the property, then one gets more examples, these are called "correct classes".

Bumby, R. T. "Modules which are isomorphic to submodules of each other." Arch. Math. (Basel) 16 (1965) 184–185. MR184973 DOI:10.1007/BF01220018

Wisbauer, Robert. "Correct classes of modules." Algebra Discrete Math. 2004, no. 4, 106–118. MR2148720 URL:author's preprint.

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