**EDIT**: I typed this up before everyone piled on to say that this is too elementary. Moving to CW rather than deleting, as I distinctly remember *not* seeing this during my education on groups and rings (but then maybe I missed class that day). And don't even *go* down the line of "grad school courses"...

Feel free to downvote if you feel like it, of course.

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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](http://mathoverflow.net/questions/3405/is-a-subgroup-of-a-free-abelian-group-free-abelian). (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.