I am reading a text where I have trouble understanding an argument:
Let $f: \mathbb C^*\times \mathbb C^*\to \mathbb C^*\times \mathbb C^*$ an isomorphism, such that $f(\{1\}\times \mathbb C^*)= \{1\}\times \mathbb C^*$.
Further, let $|x|\neq 1$.
The following argument is made:
$f_{\{1\}\times \mathbb C^*}: \{1\}\times \mathbb C^*\to \{1\}\times \mathbb C^*$ is an isomorphism of linear groups. Therefore $f(1,|x|^2)\in \{(1,|x|^2),(1,|x|^{-2}\}$.
Why should this follow? As far as I see it, the condition $f(\{1\}\times \mathbb C^*)= \{1\}\times \mathbb C^*$ means that $f_{\{1\}\times \mathbb C^*}$ comes from an isomorphism $\mathbb C^* \to \mathbb C^*$. But these could map $|x|^2$ to any value. For example the homomorphism could be a rotation.