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.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 6$: Isomorphisms of Algebraic Structures: Exercise $6.9$