Additive Group of Real Numbers is Not Isomorphic to Multiplicative Group of Real Numbers/Proof 3

Proof
There are two element of $\struct {\R_{\ne 0}, \times}$ which are self-inverse, that is, $-1$ and $1$.

However, there is only one element of $\struct {\R, +}$ which is self-inverse, that is, $0$.

there exists an isomorphism $f: \struct {\R_{\ne 0}, \times} \to \struct {\R, +}$.

Then:

So we have demonstrated that $f$ is not an injection.

Hence $f$ is not a bijection and so not an isomorphism.

It follows from Proof by Contradiction that there can be no such isomorphism.

Hence the result.