# Is there any isomorphism between the non-zero complex numbers under multiplication and the complex numbers under addition? [duplicate]

It seems like there is no trivial isomorphism between them and both are abelian and non-cyclic, so, it makes it even harder to conclude anything. I need some hints.

If I had to make a guess, I'd probably say there is no isomorphism between them.

## marked as duplicate by Arnaud D., Cam McLeman, Delta-u, Mostafa Ayaz, StrantsSep 19 '18 at 16:02

• There is one and only one solution to $nx=a$ for $a\in\mathbb{C}$, $n>0$; however, there are $n$ solutions to $x^n=a$ for $a\in\mathbb{C}$. – Arturo Magidin Sep 18 '18 at 19:07