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

Theorem
Let $\struct {\C_{\ne 0}, \times}$ be the multiplicative group of complex numbers.

Let $\struct {\C, +}$ be the additive group of complex numbers.

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

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