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So if a Lie group $G$ acts transitively on a manifold $X$, then $X$ is diffeomorphic to the quotient $G/H$ where $H$ is the stabilizer subgroup of a point $x$ in $X$.

Now if $N$ is a normal subgroup of $G$, then $N\cdot x\cong N/(N\cap H)$.

Can we imitate the second isomorphism theorem, i.e, is it true that $N/(N\cap H) \cong NH/H$ as manifolds?

Ronald
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    Not necessarily. See for example https://math.stackexchange.com/questions/448303/a-counter-example-of-the-second-isomorphism-theorem-for-topological-groups – Maik Pickl Aug 21 '18 at 09:11

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