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.