# Error in my book

1. Feb 3, 2008

### ehrenfest

[SOLVED] error in my book

1. The problem statement, all variables and given/known data
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?

2. Relevant equations

3. The attempt at a solution

2. Feb 3, 2008

### StatusX

I think the books right. Can you spell out your reasoning a little more?

3. Feb 3, 2008

### ehrenfest

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. Feb 3, 2008

### Dick

Hmm. And I think it's just Z_2. I mean, it has to be a finite group doesn't it?

5. Feb 3, 2008

### StatusX

Construct a map $\phi:\mathbb{Z} \times \mathbb{Z} \rightarrow \mathbb{Z}$ by $\phi(a,b)=2a-b$. It's easy to check that this is a surjective homomorphism, and its kernel is <(1,2)>, so by the isomorphism theorem:

$$\mathbb{Z} \times \mathbb{Z}/<(1,2)> \cong\mathbb{Z}$$

6. Feb 3, 2008

### Dick

Bing! Sure. It's not (Z/Z)x(Z/(2*Z)). Thanks, StatusX.