So I have been given these definitions:
-A G-structure on a manifold M is a principal G-subbundle P of F(TM).
-A G-structure is called integrable if the coordinate frames are in P.
and from there I am trying to show what is supposed to be a trivial example, every SL(n)-structure is integrable, but I can't manage to do it. My idea was to pick a random chart $(U,\varphi)$ and then try to make $$\frac{\partial}{\partial\varphi_1},\ldots,\frac{\partial}{\partial\varphi_n}$$ belong to $P$ by composing with some linear map $L:\mathbb{R}^n\rightarrow\mathbb{R}^n$, to get a new chart $(U,\psi)$ where $\psi=\varphi\circ L$. The problem I find with this method is that I think I can only make it happen for one point, instead of the whole $U$, any suggestions?
EDIT: Any reference where I can read about this would be much appreciated, since I can't seem to find many.