Is there a isomorphism between N and Q?

[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!

 

[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?

 

[Hurkyl]

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.

 

[JasonRox]

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

It's a step in the right direction though.

 

[twoflower]

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