Error in My Book: Z cross Z/<(1,2)> = Z_2

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In summary, there was a discussion about an error in a book regarding the equation (Z cross Z)/<(1,2)> = Z. One person argued that it equals Z cross Z_2, while the other person believed it to be just Z_2. The reasoning behind this was discussed, with one person mentioning a map and the isomorphism theorem. Ultimately, it was determined that the correct group is Z and not (Z/Z)x(Z/(2*Z)).
  • #1
ehrenfest
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[SOLVED] error in my book

Homework Statement


My book says that (Z cross Z)/<(1,2)> = Z. I say it equals Z cross Z_2. This is easy to see if you draw it out on a lattice plane. Right?


Homework Equations





The Attempt at a Solution

 
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  • #2
I think the books right. Can you spell out your reasoning a little more?
 
  • #3
You can choose any lattice point on the line y=0 or the line y=1 and get a unique line with slope 2 that goes through that point.
 
  • #4
Hmm. And I think it's just Z_2. I mean, it has to be a finite group doesn't it?
 
  • #5
Construct a map [itex]\phi:\mathbb{Z} \times \mathbb{Z} \rightarrow \mathbb{Z}[/itex] by [itex]\phi(a,b)=2a-b[/itex]. It's easy to check that this is a surjective homomorphism, and its kernel is <(1,2)>, so by the isomorphism theorem:

[tex]\mathbb{Z} \times \mathbb{Z}/<(1,2)> \cong\mathbb{Z}[/tex]
 
  • #6
Bing! Sure. It's not (Z/Z)x(Z/(2*Z)). Thanks, StatusX.
 

Related to Error in My Book: Z cross Z/<(1,2)> = Z_2

1. What is the meaning of "Z cross Z/<(1,2)> = Z_2" in your book?

"Z cross Z/<(1,2)> = Z_2" is a mathematical equation that represents a group structure known as the cyclic group of order 2. It is a group of integers modulo 2, where the elements can only take on two values - 0 or 1.

2. How did you come up with this equation?

This equation is a well-known result in abstract algebra that is often used to illustrate the concept of quotient groups. It is a fundamental concept in modern algebra and can be found in many textbooks and research papers.

3. Can you explain the significance of Z cross Z/<(1,2)> = Z_2 in real-world applications?

This equation has many applications in various fields such as coding theory, cryptography, and group theory. In coding theory, it is used to create error-correcting codes to ensure accurate transmission of data. In cryptography, it is used to generate keys for secure communication. And in group theory, it is used to study the structure and properties of different groups.

4. What are the common misconceptions about this equation?

One common misconception is that Z cross Z/<(1,2)> = Z_2 means that 1 and 2 are the only elements in the group. However, this equation only represents the order of the group, and there can be many other elements in the group. Another misconception is that Z_2 is the same as the set {0,1}, but in reality, Z_2 is a group with specific operations defined on it.

5. How can readers understand and apply this equation in their own work or research?

Readers can understand and apply this equation by first familiarizing themselves with the basics of group theory and quotient groups. They can then use this equation to solve problems in their own work or research that involve group structures or modular arithmetic. It is also essential to understand the limitations and assumptions of using this equation and to consult with experts if needed.

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