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Fix integers a,b,c,d.

Let H be the subgroup of ZxZ generated by (a,b) and (c,d).

When (in terms of a,b,c,d) is the factor group (ZxZ)/H finite?

I figured that if ad is not equal to bc then the factor group (ZxZ)/H is of order ad-bc, and if ad is equal to bc then the factor group (ZxZ)/H is infinite. But I can only figure out how to prove it by drawing diagrams and showing it geometrically. Are there any rigorous ways to prove that?

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# Factor group

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