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.

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.

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.

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.

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.

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.

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.