Every fiber bundle has the group of homeomorphisms from the fiber to itself as structure group. However, this is a very huge group. I was wondering whether the structure group can always be reduced to at least something finite-dimensional (for the fiber being a finite-dimensional manifold)?
If I remember correctly, I read somewhere that $U(1)$ principle bundles are equivalent to oriented circle bundles. If that's true, then for circle bundles we can always reduce from the huge group of circle automorphisms to $U(1)$ (guess for the unoriented case something like $U(1)\rtimes \mathbb{Z}_2$ will do). For other Lie groups, is it true that any fiber bundle with the space of the Lie group as fiber admits that Lie group as structure group? More generally, for any fixed finite-dimensional manifold as fiber, is there a finite-dimensional group (with action on the fiber) which can always be used as structure group?