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

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

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

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

Proof
$\struct {\C_{\ne 0}, \times}$ is isomorphic to $\struct {\R, +}$.

Let $\phi: \C_{\ne 0} \to \R$ be an isomorphism.

Note that $\order i = 4$ in $\struct {\C_{\ne 0}, \times}$.

By Order of Isomorphic Image of Group Element:
 * $\order {\map \phi i} = 4$ in $\struct {\R, +}$

Then $4 \map \phi i = 0$.

So $\map \phi i = 0$.

However $\order 0 = 1$.

Thus there is no order $4$ element in $\struct {\R, +}$, and thus $\map \phi i$ does not exist.

Therefore $\phi$ is not an isomorphism.

Hence the result by Proof by Contradiction.