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proposition : Let $G$ be a Lie group. Then $G$ as a topological group admits, up to isomorphism, only one differentiable structure turning it into a Lie group.

I think I have to use the fact that if $f : G\to H$ is a continuous homomorphism between Lie groups then $f$ is smooth. In particular, I should apply this to $\operatorname{Id}_G : G\to G$ but I am not sure how this gives us the result.

roi_saumon
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  • This question might be of interest. – Kajelad Aug 19 '20 at 21:49
  • The trick is to spell out the exact meaning of your question: You have to show that if $G_1, G_2$ are two (smooth) Lie groups and $\phi: G_1\to G_2$ is a homeomorphic isomorphism, then $\phi$ is a diffeomorphism. Is $\phi$ is smooth? What about $\phi^{-1}$? – Moishe Kohan Aug 20 '20 at 16:09

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