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In Glen Bredon "Topology and Geometry", Ch. II-13, I am stuck on the following

"Problem 1. Finish the proof at the end of this section that $G\rightarrow G/H$ is a bundle. Also show that this is a fiber bundle with structure group $N(H)/H$ where $N(H)$ is the normalizer of $H$ in $G$."

Here $H$ is a closed subgroup of the Lie group $G$ (which the book assumes to be again a closed subgroup of $Gl(n, \mathbb{C})$."

I was able to compute that this is a fiber bundle with structure group $H$, using the trivializations given in the text. Obviously, some other trivializations of the same bundle will be required to get $N(H)/H$ as structure group. I have now been stuck on this for a while, and any help would be greatly appreciated.

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