Additive Group of Rational Numbers is not Isomorphic to Multiplicative Group of Rational Numbers

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Theorem

Let $\struct {\Q, +}$ be the additive group of rational numbers.

Let $\struct {\Q_{\ne 0}, \times}$ be the multiplicative group of rational numbers.


Then $\struct {\Q_{\ne 0}, \times}$ is not isomorphic to $\struct {\Q, +}$.


Proof

A direct application of Additive Group and Multiplicative Group of Field are not Isomorphic.

$\blacksquare$


Sources