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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.

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

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  • $\begingroup$ 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 $a\in\mathbb{C}$. $\endgroup$ – Arturo Magidin Sep 18 '18 at 19:07
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There is no isomorphism between them, and the reason is very simple. In the group of the non zero complex numbers under multiplication there are a lot of non trivial elements of finite order. (think about the roots of unity). On the other hand in the group of complex numbers under addition every non trivial element has infinite order. Isomorphism preserves order of elements, so they can't be isomorphic.

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