Let $C(r)$ be the group of $r-$roots of unity. I read that $C(r)$ acts on $\mathbb C$ as a reflection group. Could you please explain how is that?
Here is my understanding of the action: $C(r)=<\alpha>$, for some primitive $r-$root of unity $\alpha\in \mathbb C$. The action is defined by the multiplication: $C(r)\times \mathbb C \longrightarrow \mathbb C$, $(\alpha, z) \longmapsto \alpha z$.
The definitions I have in mind are the following: A reflection is a finite order automorphism $f\in GL(\mathbb C)$ such that $\ker(f-id)$ is a hyperplane of $\mathbb C$.
A reflection group is any subgroup of $GL(\mathbb C)$ generated by reflections.
A complex reflection of $\mathbb C^n$ is an element $f\in GL_n(\mathbb C)$ of finite order such that $\ker(f-id)$ is of complex dimension $n-1$. Thus, a complex reflection of $\mathbb C$ is a root of unity distinct from $1$. $C(r)$ is generated by such an element. According to en.wikipedia.org/wiki/Complex_reflection_group, $C(r)=G(r,1,1)$.