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

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.