Additive Group of Real Numbers is Not Isomorphic to Multiplicative Group of Real Numbers/Proof 2

Theorem

Let $\struct {\R, +}$ denote the additive group of real numbers.

Let $\struct {\R_{\ne 0}, \times}$ denote the multiplicative group of real numbers.

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

Proof

From Real Numbers form Field, $\struct {\R, +, \times}$ forms a field.

The result then follows as an example of Additive Group and Multiplicative Group of Field are not Isomorphic.

$\blacksquare$