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

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In summary, the conversation discusses the isomorphism between ZxZ/<(a,b)> and Z, where gcd(a,b)=1. The attempt at a solution involves a geometric approach, graphing points on a line in the x-y plane and shifting it along the y-axis to hit all points. However, for ZxZ/<(2,3)>, the cosets of (1,y)+<(2,3)> must also be considered, leading to the conclusion that ZxZ/<(2,3)> is isomorphic to Z2xZ. The speaker requests clarification on this solution.
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y2kevin
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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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any ideas?
 

1. What is a show isomorphism?

A show isomorphism is a mathematical concept that refers to a mapping between two algebraic structures that preserves their structure and operations. In other words, it is a way of demonstrating that two structures are essentially the same.

2. What is ZxZ/<(a,b)>?

ZxZ/<(a,b)> is a mathematical notation that represents the set of all ordered pairs of integers, ZxZ, with the relation <(a,b)> defined as the equivalence relation that identifies elements that are multiples of (a,b). In other words, it is the set of all possible remainders when dividing the integers by (a,b).

3. How is the isomorphism of ZxZ/<(a,b)> to Z related to gcd(a,b)=1?

The isomorphism of ZxZ/<(a,b)> to Z is related to gcd(a,b)=1 because when the greatest common divisor of a and b is equal to 1, it means that there are no common factors between a and b. This results in the equivalence relation <(a,b)> only identifying multiples of 1, which is equivalent to saying that all integers in ZxZ/<(a,b)> are distinct. Therefore, ZxZ/<(a,b)> is isomorphic to Z, as both sets have the same number of elements.

4. How can the isomorphism of ZxZ/<(a,b)> to Z be shown?

The isomorphism of ZxZ/<(a,b)> to Z can be shown by constructing a specific mapping, known as an isomorphism, between the two sets. This mapping should preserve the structure and operations of both sets, meaning that it should be a one-to-one and onto, as well as maintain the algebraic properties of addition and multiplication.

5. What is the significance of showing the isomorphism of ZxZ/<(a,b)> to Z?

The significance of showing the isomorphism of ZxZ/<(a,b)> to Z is that it provides a way to understand the structure of ZxZ/<(a,b)>, which may be a complex or unfamiliar algebraic structure, by relating it to a more familiar and simpler structure, Z. This can aid in solving problems and making connections between different mathematical concepts.

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