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Is there a isomorphism between N and Q?

  1. Dec 20, 2006 #1
    Hi all,

    I wonder if there is an isomorphism between the group of [itex]\mathbb{N}[/itex] and the group of [itex]\mathbb{Q}[/itex] (or [itex]\mathbb{Q}+[/itex]). 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. jcsd
  3. Dec 20, 2006 #2


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    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
  4. Dec 20, 2006 #3


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    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.
  5. Dec 20, 2006 #4


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    It's a step in the right direction though.
  6. Dec 20, 2006 #5
    Thank you, I've got it, I mixed the general set bijection and isomorphism between algebraic structures in my mind.
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