Are $\mathbb{C}$ under addition and $\mathbb{C}$ under multiplication isomorphic?

Question

Is it true that $\mathbb{C}$ under addition and $\mathbb{C}^*$ under multiplication isomorphic?

I don't think we can use the argument that, $\mathbb{C}$ have zero inside it, but, $\mathbb{C}^*$ don't have. As, they are infinite sets anything can happen.

I have proved that, $(\mathbb{R},+)$ and $(\mathbb{R}^+,\cdot)$ are isomorphic using the map $x\mapsto e^x$. Can I use that? I can't see how that map will work here.

Can anyone please explain hoe to proceed? Any hint will be appreciated.

Answer 1

No, because their torsion subgroups are different.

Indeed, $(\mathbb C^*,\times)$ has an infinite number of torsion elements (the roots of unity), but $(\mathbb C,+)$ has just one torsion element (zero).

(A torsion element is an element of finite order.)

Therefore, no group homomorphism $(\mathbb C^*,\times) \to (\mathbb C,+)$ can be injective.

Answer 2

You do not specify what sort of automprphism is wanted. I will show $(\mathbb C,+)$ and $(\mathbb C^*,\cdot)$ are not isomorphic as groups. The group $(\mathbb C,+)$ has no element of order $2$. [$z+z=0$ implies $z=0$.] But the group $(\mathbb C^*,\cdot)$ does, namely $-1$.