I am learning group theory on my own so I may not know the correct way of thinking in a branch of math thats completely foreign in terms of the style of proofs. How to prove that the multiplicative groups $(\mathbb R -\{0\})^{\times}$ and $(\mathbb C -\{0\})^{\times}$ are not isomorphic?
I tried to identify that $(\mathbb C -\{0\})^{\times}$ consists of $\{x+yi | x,y \in \mathbb R, x,y\neq 0\}$, but I don't know how to prove rigorously that we cannot find an isomorphism or homomorphism $\phi$ such that $\phi(xy)=\phi(x)\phi(y)$. It just seems trivial but I don't know how to prove it convlusively. Thanks!
Hint: $\mathbb{R}^{\times}$ has no elements of order $n$ for any integer $n>2$. What about $\mathbb{C}^{\times}$?