Let $M$ be a smooth manifold and let $G$ be a finite group acting on $M$ by diffeomorphisms. Show that the set of fixed points $M^G := \{m \in M \mid g . m = m, \forall g \in G\}$ is a smooth manifold
What I tried
Since $M$ is a smooth manifold, then there exist a maximal smooth atlas $\mathfrak{A}=\{(U_\alpha, V_\alpha, \phi_\alpha)\}_{\alpha \in I}$. Now I want to prove that the fixed point set $M^G$ defined above is a smooth manifold and I want to establish a chart for the same. I am unable to understand how will I use the fact that the group $G$ is finite here. Also I think it is diffcult for me to get some charts for $M^G$. Can anyone help me this way?