# Is there a isomorphism between N and Q?

1. Dec 20, 2006

### twoflower

Hi all,

I wonder if there is an isomorphism between the group of $\mathbb{N}$ and the group of $\mathbb{Q}$ (or $\mathbb{Q}+$). I know there is a proof that there is a bijection between these sets, but I didn't find a way how to construct the isomorphism.

What confuses me a little is that (I think) the group of natural numbers has only one generator, while the group of (positive) rationals has more than one generator, so I can't see how the mapping would look like.

Thank you for any hints!

2. Dec 20, 2006

### StatusX

Well you've answered your own question, since the image of the generator should generate the image of the entire group. Can you make this into a proof?

Last edited: Dec 20, 2006
3. Dec 20, 2006

### Hurkyl

Staff Emeritus
Er, N isn't a group. Did you mean Z, with addition as the operation?

(And, I assume you meant addition for Q and multiplication for Q+)

There's no reason to think that the existence of a bijection as sets implies that there is an isomorphism as groups.

4. Dec 20, 2006

### JasonRox

It's a step in the right direction though.

5. Dec 20, 2006

### twoflower

Thank you, I've got it, I mixed the general set bijection and isomorphism between algebraic structures in my mind.