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Yemon Choi
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Interesting question! The following is not a full answer, but seemed worth a bit more than a comment.

If I understand correctly, the first part asks if G is free abelian, in which case the answer is apparently yes, see this MO question from someone reputable. (I confess this is not something I knew, although I did have my suspicions.)

My guess is that some modification of Gauss-Jordan elimination should provide an algorithm for extracting a "Z-basis" from a Z-generating set, but it's not clear to me right now how one would get a Z-generating set from a given Q-basis of your original V.

Yemon Choi
  • 25.8k
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  • 156