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Prove that $(\mathbb{Z},+) \text{ and } (\mathbb{Q},+)$ are not isomorphic.

Usually in order to show that two groups are not isomorphic I would show that either the groups have different orders, one is abelian and the other isn't, one is cyclic and the other isn't or one contains an element of order d and the other doesn't. I can see that this isn't going to work here though.

I feel like a proof by contradiction is the way to go with this, I'm not sure how to proceed with that in this case though.

rubik
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MHW
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  • There are probably more ways to do it, but I think contradiction is the way to go. You assume you have an isomorphism, clearly also surjective. Can you find a relation in Q that does not have a solution in Z? – Rico Mar 07 '16 at 19:43
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    http://math.stackexchange.com/questions/620551/prove-that-the-additive-groups-mathbbz-and-mathbbq-are-not-isomorphic – Jaakko Seppälä Mar 07 '16 at 19:47
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    Note that $(\mathbb{Z},+)$ is a cyclic group. Can you show that $(\mathbb{Q},+)$ is not a cyclic group? – Dave L. Renfro Mar 07 '16 at 19:48

3 Answers3

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Hint: $$\forall q\in\mathbb{Q} \exists p : p+p=q$$

Uri Goren
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Uri Goren has the right of it, but I thought it worth elaborating a bit if you're still getting the hang of this sort of argument.

As you've essentially noticed already, you can prove this by observing that $(\mathbb{Q},+)$ has a "group theoretic property" that $(\mathbb{Z},+)$ does not. In other words, identify a statement about $(\mathbb{Q},+)$ that can't possibly be preserved by an isomorphism $\varphi: \mathbb{Q} \to \mathbb{Z}$ of additive groups. Formalizing such a proof is often done either (a) by contradiction or (b) by showing no homomorphism $\varphi: \mathbb{Q} \to \mathbb{Z}$ exists that is surjective (or injective).

For instance, suppose $\varphi: (\mathbb{Q},+) \to (\mathbb{Z},+)$ is a homomorphism. Then $\varphi(1)$ is an integer, as is $\varphi(1/n)$ for all $n \geq 1$, and you can show that

$$ \varphi(1/n) = \varphi(1)/n. $$

Thus $\varphi(1)/n \in \mathbb{Z}$ for all $n \in \mathbb{N}$, so what is $\varphi(1)$? What can you conclude about $\varphi(1/n)$ and $\varphi(m/n)$ accordingly?

Dan
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$(\mathbb{Z},+)$ is cyclic whereas $(\mathbb{Q},+)$ is not cyclic.

InsideOut
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