Proof regarding direct sum of 2 copies of (Q,+)
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Use the fact that any finitely generated subgroup of (Q,+), (the rationals under addition), is cyclic to show that (Q,+)+(Q,+), (the direct sum of two copies of (Q,+)), is not isomorphic to (Q,+). The first part of this was showing that any finitely generated subgroup of (Q,+) is cyclic and I understand that part and really had little difficulty with it, but this second part is proving far more difficult. Any help would be greatly appreciated.
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This solution is comprised of a detailed explanation to proof regarding direct sum of 2 copies of (Q,+).
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(In what follows, the group operation is addition throughout)
Consider the subgroup of generated by the elements and (this is just )
The cyclic subgroup generated ...
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