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

[ehrenfest]

[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

 

[StatusX]

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

 

[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.

 

[Dick]

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

 

[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:

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

 

[Dick]

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