Suppose that $G$ is a Lie group with a transitive action on a smooth manifold $M$. The regular theory of Lie groups tells us that $G$ and $M$ are diffeomorphic if the isotropy group is trivial.
The proof that I know goes, as usual, by proving that the canonical map $G/H \rightarrow M$ is bijective and its differential is injective. This implies that the canonical map is a diffeomorphism, but heavily relies on the fact that both manifolds are finite dimensional.
Suppose now that both $G$ and $M$ are infinite dimensional manifolds. Is it still true that $G$ and $M$ are diffeomorphic if the isotropy group is trivial? Or asked differently, is there a direct proof that the differential of $G/H \rightarrow M$ is surjective?
I just realized that the proof I know also depends on the SMOOTH exponential map, which can be defined in the infinite dimensional case, but according to "INFINITE DIMENSIONAL LIE GROUPS WITH APPLICATIONS TO MATHEMATICAL PHYSICS" by Rudoph Schmid is not always differential (the counterexample is the group of $s$-Sobolev diffeomorphisms).