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$\textbf{Question: }$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.

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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.

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