I'm currently doing my master in Math and I've been reading Milnor-Stasheff's Characteristic Classes.
After introducing vector bundles there is a sentence that goes like this:
"An $\mathbb{R}^n$-bundle is a fiber bundle with fiber $\mathbb{R}^n$ and with the full linear group GL$_n(\mathbb{R})$ as a structural group."
My question is what does it mean to be a structural group?
In vector bundles, we have (the local triviality property) for every point $b$ of base space $B$ a neighbourhood $U_b$, an integer $n$ and a homeomorphism
$$h:U_b \times\mathbb{R}^n\to \pi^{-1}(U_b,)$$ where $\pi:E\to B$ is the projection map from the total space $E$ to the base space $B$ of a vector bundle. Then, if I got this right, we can observe this as a map $$h_b:\mathbb{R}^n\to \pi^{-1}(b)$$ which is an isomorphism (with $x\mapsto h(b,x)$). So fiber $F_b=\pi^{-1}(b)$ is isomorphic to $\mathbb{R}^n$.
If the dimension $n$ of the fibers (which can be a function) is a constant, then we talk about $\mathbb{R}^n$-bundle. Now I really don't get what does it mean that this has GL$_n(\mathbb{R})$ as a structural group. What does structural group mean?
I hope I managed to explain what I do not get, and it is possible that I understood some parts wrong. Thank you in advance for possible clarifications!
In the picture of local trivializations of a vector bundle, you also have a condition on the relation between different trivialzations. Usually, this takes the form that (the appropriate restrcition of) $h^{-1}\circ \tilde h$ which maps $(U_b\cap U_{\tilde b})\times\mathbb R^n$ to itself is linear in the second varaible. (This is needed for the fibers of the bundle to inherit a canonical structure of a vector space, otherwise different trivializations would lead to different addition and/or scalar multiplication on the fibers.) Without this condition, you would just have a fiber bundle with fibers that are homeomorphic to $\mathbb R^n$.
The condition of linearty in the second variable can be equivalently phrased as the fact that the chart chage has the form $ h^{-1}(h(x,v))=(x,A(x)v)$ for a continuous function $A$ from $U_b\cap U_{\tilde b}$ to the group $GL(n,\mathbb R)$ of invertible $n\times n$-matrices. Therefore, it is also phrased as the fact that the bundle (or better the bundle-atlas) in question has structure group $GL(n,\mathbb R)$.