# Abstract group acting on a smooth manifold

$$\textbf{Question: }$$Let $$\Gamma$$ be an abstract group acting as a diffeomorphisms on a $$C^\infty$$ manifold $$M$$ and assume that the isotropy group $$\Gamma_p$$ of $$p$$ is finite for every $$p\in M$$. If the orbits are closed, discrete subsets of $$M$$, then prove that $$\Gamma$$ is necessarily countable and the action is discontinuous.

Proving that the action is discontinuous was not so difficult, but couldn't prove that $$\Gamma$$ is countable. Please help.

Remember your manifold $$M$$ is second-countable. Can you see why the discrete orbit $$\Gamma\cdot p$$ must be countable? Combine that with $$\Gamma_p$$ is finite.