I have to show whether the groups $(\mathbb{C}, +, 0)$ and $(\mathbb{C} − \{0\}, \times, 1)$ are isomorphic.
I know that two groups are isomorphic if there exists an isomorphism from one to the other. An isomorphism from $G_1$ to $G_2$ is a bijective homomorphism. We call $G_1$ and $G_2$ isomorphic, and write $G_1 \cong G_2$ if an isomorphism from $G_1$ to $G_2$ exists. I tried the following by using the lecture notes:
$\varphi:\mathbb{C}− \{0\} \mapsto \mathbb{C}$, then $x\in \mathbb{C}−\{0\}$, and $\varphi(x)\in \mathbb{C}$, then $|x|=|\varphi(x)|$. Now note that the element $i\in \mathbb{C}−\{0\}$ has order $4$. However, no element in $\mathbb{C}$ has order $4$, I think???. Hence, no isomorphism can exist. Hence, the groups $\mathbb{C}−\{0\}$ and $\mathbb{C}$ are not isomorphic.
Any tips or help would be grateful!