There is this question in my book and I'm not sure what the geometric meaning of this situation is.
Consider the unit circle in the complex plane $$S^1=\{z\in\mathbb{C}:|z|=1\}.$$ Let $n$ be a positive integer, consider the $n$-th root of unity $$\xi=\cos\left(\frac{2\pi}{n}\right)+i\sin\left(\frac{2\pi}{n}\right)\in\mathbb{C}$$ and let $\mathbb{Z}_n$ be the (additive) group of integers modulo $n$. Show that $$\phi_k(z):=\xi^kz$$ defines an action of $\mathbb{Z}_n$ on $S^1$, explain its geometric meaning and show that $S^1/\mathbb{Z}_n$ is homeomorphic to $S^1$. What do you obtain when $n=2$?
So, a bunch of questions, but firstly I'm interested in the geometric meaning. What does this situation mean? With this knowledge I hope to be able to answer the other questions of the exercise.
Thanks in advance!
Note that $\xi = e^{\frac{2\pi i}{n}}$, so, due to how nultiplication of complex numbers works, multiplication by $\xi$ is a counterclockwise rotation by that angle of $\frac{2\pi}{n}$.
You can express this in matrix form. Let $\mathbb C \cong \mathbb R^2$ as real vector spaces, and pick a basis $\{1, i\}$. Then, the operator of multiplication by $\xi$ looks like
$$\begin{pmatrix}\cos\frac{2\pi}{n} & \sin\frac{2\pi}{n} \\ -\sin\frac{2\pi}{n} & \cos\frac{2\pi}{n}\end{pmatrix}$$
which is exactly the matrix of the plane rotation by the angle $\frac{2\pi}{n}$.