Quick question: 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?