Isomorphism between $\mathbb{C}^* $ under multiplication with $\mathbb{C}$ under addition.

Does there exists an Isomorphism between $\mathbb{C}^* $ under multiplication with $\mathbb{C}$ under addition.

Attempt In my opinion It does not exists.

If there exists such an isomorphism say $\phi :\mathbb{C}^* \to \mathbb{C}$ then, from the fact that Isomorphism takes identity to identity we have $$\phi \left( 1 \right)=0$$

From the fact

Given $\phi:G \to H$ be an isomorphism, then For a fixed integer $k$ and a fixed group element $b$ in $G$, the equation $x^k=b$ has the same number of solutions in $G$ as does the equation $x^k=\phi\left(b\right)$ in $H$.

We have $x^4=1$ and $x^4=0$ must have same number of solutions, but the solutions to these equations are 4 and 1 only.

Solution 1:

No, no such isomorphism exists.

For if

$\phi: \Bbb C^\ast \to \Bbb C \tag 1$

were such an isomorphism, it would, as pointed out by our OP Rakesh Bhatt, satisfy

$\phi(1) = 0; \tag 2$

one may argue this in the usual way, via the fact that group isomorphisms take identities to identities, or even more generally by observing that

$\phi(1) = \phi(1^2) = \phi(1) + \phi(1), \tag 3$

whence

$\phi(1) = 0. \tag 4$

In any event,we may use (2), (4) to resolve our problem as follows: let $1 \ne \omega \in \Bbb C^\ast$ be an $n$-th root of unity where $n > 1$;. then

$\omega^n = 1, \tag 5$

whence

$\phi(\omega^n) = \phi(1) = 0; \tag 6$

but

$\phi(\omega^n) = \phi(\omega) + \phi(\omega) + \ldots + \phi(\omega), \; n \; \text{times} = n \phi(\omega); \tag 7$

it follows from (6), (7) that

$n \phi(\omega) = 0, \tag 8$

whence

$\phi(\omega) = 0; \tag 9$

since $\phi$ is an isomorphism, we now conclude from (2) and (9) that

$\omega = 1, \tag{10}$

contrary to our choice of $\omega$; thus no such isomorphism $\phi$ can exist.