Let $G$ be a compact Lie group acting freely by diffeomorphisms on a compact smooth manifold $M$. Prove that the quotient $M/G$ is also a smooth manifold.
I can prove that the equivalence relation $x\sim y \Leftrightarrow gx=y$ for some $g\in G$ is open (since the action is by diffeomorphisms), so $M/G$ is second countable. I can also prove that $G$ compact $\Rightarrow M/G$ Hausdorff.
But I'm having trouble finding a differentiable structure for $M/G$. I tried to use the charts $\psi: U\to V\subset\mathbb{R}^n$ from the manifold $M$ to form charts $\bar{\psi}:[U]\to \bar{V}\subset\mathbb{R}^m$, but I don't know $m$.
Any ideas? Thanks!