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Let $G$ be a Lie group and $$G_x:=\{g \in G; Ad^*_g(x)=x\}$$ the stabilizer, where $Ad_g^*$ is the adjoint of the adjoint representation.

Now, I was wondering why $G/G_x$ has a manifold structure. Afaik, the criterion for a quotient Lie group to be again a manifold is that the stabilizer group here is a normal subgroup. Unfortunately, I don't see why this should be the case.

Could anybody give me a hint, why $G/G_x$ is still a manifold?

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For putting a manifold structure on the quotient, it's irrelevant whether $G_x$ is normal. The quotient of a Lie group modulo a closed subgroup always has a unique smooth manifold structure such that the group action is smooth -- this is the Quotient Manifold Theorem (see Theorem 21.10 in my Introduction to Smooth Manifolds, 2nd ed.). Normality of the subgroup is necessary in order for the quotient to be a Lie group.

If a Lie group $G$ acts smoothly (or even continuously) on a manifold $M$, then every stabilizer group $G_x$ is automatically closed, so $G/G_x$ is always a smooth manifold.

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