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.© BrainMass Inc. brainmass.com October 9, 2019, 10:32 pm ad1c9bdddf
(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 ...
This solution is comprised of a detailed explanation to proof regarding direct sum of 2 copies of (Q,+).