Let $G$ be a lie group, $H\subseteq K\subseteq G$ be closed subgroups , and $H$ be normal in $G$. I wonder if the coset manifold $\frac{\frac{G}{H}}{\frac{K}{H}}$ is diffeomorphic to $\frac{G}{K}$.
I would be very thankful for any help.
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$\begingroup$ Yes, and the map is $gH \mapsto (gH)(K/H)$. Check that it descends to a map $G/K \to (G/H)/(K/H)$ and is 1-1. Because it is $G$-equivariant, it is a smooth map of homogeneous spaces, and a smooth fiber bundle map. Since it is 1-1, it is a diffeomorphism. $\endgroup$– Ben McKayCommented Feb 4, 2014 at 9:50
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$\begingroup$ For a more general statement, see Bourbaki, Lie groups and Lie algebras, Chapter 3, §1, Proposition 13. $\endgroup$– abxCommented Feb 4, 2014 at 10:08
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