Show Isomorphism of ZxZ/<(a,b)> to Z if gcd(a,b)=1

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SUMMARY

The discussion centers on proving that the quotient group ZxZ/<(a,b)> is isomorphic to Z when gcd(a,b)=1. The initial reasoning involved geometric interpretations of cosets in the Cartesian plane, specifically for ZxZ/<(1,a)>, which successfully demonstrated the isomorphism to Z. However, the attempt to apply similar reasoning to ZxZ/<(2,3)> led to confusion, as it was incorrectly concluded that it is isomorphic to Z2xZ due to missing cosets. The correct understanding hinges on the properties of cosets and the implications of the gcd condition.

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Homework Statement


Show that ZxZ/<(a,b)> is isomorphic to Z if gcd(a,b)=1.


The Attempt at a Solution



I thought I had an idea but apparently I don't.

I reasoned this geometrically. For ZxZ/<(1,a)> (for all a in Z) can be graphed as a line hitting points (k,a*k) in the x-y plane. If we shift the line covered by <(1,a)> along the y-axis (e.g. use cosets (0,y)+<(1,a)>, where y is an integer), we can hit all points in ZxZ. Hence, ZxZ/<(1,a)> is isomorphic to Z.

But for ZxZ/<(2,3)>, the story changes, we skip all the (1,y) values if we shift by (0,y)+<(2,3)>. Hence, this leads me to conclude that we also need to include the possible cosets of (1,y)+<(2,a)>, making ZxZ/<(2,3)> isomorphic to Z2xZ. But apparently this is wrong.

Can anyone shed a light on this? Thank you.
 
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