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Suppose $G$ is a Lie group acting on the manifold $M$ and $p \in M$ is such that $G_p$, the isotropy group of $p \in M$, is closed in $G$. I'm trying to prove that $G/G_p$ has a manifold structure.

To do this, I defined $F: G/ G_p \rightarrow M$ by $F(aG_p) = a\cdot p$ and considered the inverse image topology on $G/G_p$. Then, I took an atlas $\{U_\alpha,\phi_\alpha\}_{\alpha \in A}$ for $M$ and tried to prove that $\{F^{-1}(U_\alpha),F^*\phi_\alpha\}_{\alpha \in A}$ is an atlas for $G/G_p$. However, I'm not being able to prove that $F$ is a homeomorphism between $G/G_p$ and $M$.

Is it true that $F$ is a homeomorphism or should the topology of $G/G_p$ be defined in another way?

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