Here's an exercise given during a course in Differential Geometry that I'm taking.
Let $M$ denote a smooth manifold and let $G$ be a finite group of diffeomorphisms acting on it without fixed points (that is, $g(p)=p$ for some $p\in M$ forces $g$ to be the identity). We then have on the quotient space $M/G$ a differentiable structure. An atlas for it is obtained via the following observation: if $[p]$ is a point in $M/G$ then there exist a representative $p\in M$ and a chart $(U, \phi)$ in $p$ such that the projection $\pi\colon M\to M/G$ is injective on $U$. It then makes sense to define a chart $(\overline{U}, \phi_{\overline{U}})=\left(\pi(U), \phi\circ\left(\pi|_{U}\right)^{-1}\right).$ The family of all such charts is an atlas for $M/G$.
Exercise Show that the transition functions $\phi_{\overline{V}}\circ\phi_{\overline{U}}^{-1}$ can be identified with elements of $G$.
I find this question to be somewhat vague. Identified in which sense? The set of those transition functions might well be infinite, while $G$ is not. Also, I cannot see any relationship between the two. Can somebody provide me with some hint? Thank you.