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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.

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After a talk with the maker of the exercise I finally got it. Let us fix some notation: for every $p \in M$ let $[p]$ denote the equivalence class of $p$ in $M/G$ and let $\pi(p)=[p]$.

Fix point $x\in M$ and a diffeomorphism $g \in G$. Let $\tilde{x}=gx$. Since $G$ is finite, we can take charts $(U, \phi)$ and $(\tilde{U}, \tilde{\phi})$ in $x$ and $\tilde{x}$ respectively such that $\left.\pi\right\rvert_U$ and $\left.\pi\right\rvert_{\tilde{U}}$ are injective. This gives us two charts in quotient space: namely, if $V=\pi(U)$ and $\tilde{V}=\pi(\tilde{U})$, we have the charts $(V, \psi=\phi\circ\left(\left.\pi\right\rvert_U\right)^{-1})$ and $(\tilde{V}, \tilde{\psi}=\tilde{\phi}\circ\left(\left.\pi\right\rvert_\tilde{U}\right)^{-1})$.

The transition function relative to those two charts is related to $g$. Indeed, if $\eta\in \psi(V)\cap\tilde{\psi}(\tilde{V})\subset \mathbb{R}^n$, then \begin{align} (\psi\circ\tilde{\psi}^{-1})(\eta)&=\phi\left( \pi\tilde{\phi}^{-1}(\eta)\right) \\ \notag &=\phi\left(\left.\pi\right\rvert_U\right)^{-1}\left(\pi\tilde{\phi}^{-1}(\eta)\right)\\ \notag &=\left(\phi g^{-1}\tilde{\phi}^{-1}\right)(\eta). \end{align} This concludes the exercise.

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