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The additive group of rational numbers is not isomorphic to the multiplicative group of positive rational numbers. How should I show this? What structural property does the first group possess that the other does not?

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The additive group has the property that for all $w$ there exists $x$ such that $x+x=w$. The multiplicative group does not have the property that for all $w$ there exists $x$ such that $x\cdot x=w$, since $\sqrt{2}$ is irrational.

Remark: So one important structural difference is that the first group is divisible (the equation $x+x+\cdots +x=w$ has a solution for every $w$) while the second group is not divisible.

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