Determining Finiteness of (ZxZ)/H for a,b,c,d

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Discussion Overview

The discussion revolves around determining the conditions under which the factor group (ZxZ)/H is finite, where H is a subgroup generated by two vectors (a,b) and (c,d) in the integer lattice ZxZ. Participants explore both geometric and algebraic perspectives on the problem, seeking a rigorous proof of the relationship between the determinant of a matrix formed by these vectors and the finiteness of the factor group.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant suggests that the factor group (ZxZ)/H is finite if ad is not equal to bc, and infinite if ad equals bc, but seeks a rigorous proof beyond geometric intuition.
  • Another participant questions the non-rigorous nature of the initial approach and suggests that a proof can be valid regardless of whether it is geometric or algebraic, as long as it is logically sound.
  • A participant describes the subgroup H as consisting of points of the form (ma+nc, mb+nd), linking this to matrix operations and suggesting that the determinant ad-bc is relevant to the problem.
  • One participant expresses uncertainty about the rigor of their geometric description and contemplates the need for a more detailed proof regarding the relationship between the area of the parallelogram formed by the vectors and the number of elements in H.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the nature of rigor in proofs or the sufficiency of geometric arguments. There are competing views on how to approach the proof, with some favoring algebraic methods while others defend the validity of geometric reasoning.

Contextual Notes

Participants acknowledge the need for clarity in proving that the number of elements in H corresponds to the area represented by the determinant, but do not resolve the specifics of these mathematical steps.

e12514
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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?
 
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What is non-rigorous about what you did, and how can you go about correcting that?
 
Non-rigorous: drawing diagrams and describing what's happening instead of precise lines of algebraic proof.

Rigorous: perhaps some proof not via a geometric point of view, and purely by considering the group ZxZ and the subgroup H and the factor groups algebraicly? (If that's possible)
 
Firstly, it is not necessarily true that your first argument is not rigorous. Write out you 'non-rigorous' descriptions, and see if you can make them satisfy your notion of rigorous. If a proof is logically correct without any leaps of faith then it doesn't matter whether it is diagrammatic or not.

Anyway, the point is that your subgroup H is the set of all points (ax+cy, bx+dy), isn't it? Sums of multiples of (a,b) and (c,d). Now doesn't that look a lot like what happens when you apply a matrix to (x,y)? A matrix involving a,b,c,d, and isn't ad-bc very suggestive when you think about matrices?
 
Well yes. What I have is that the set of points in H are elements of the form (ma+nc, mb+nd) which is pictured geometrically by the set of points on ZxZ that lies on any multiple of the vector (a,b) or (c,d) or a combination of both. The number of elements in (ZxZ)/H is then the number of points contained inside inside one //gram plus the point on the bottom left vertex, which should be equal to its area, which is equal to the determinant of the matrix whose first column is a,b and second column is c,d , equalling (the absolute value of) ad-bc. With a diagram showing the elements of H and joining them up with parallel lines etc.

The only problem is I think that was just a description rather than a proper proof, and I probably need to go into more detail for proving certain parts, perhaps that the number of elements in H is equal to the area which is equal to the determinant... etc.
 

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