# What is the structure group of an associated bundle?

Let $$p:E\rightarrow B$$ be a principal G-bundle such that G acts on a space F on the right. The "associated locally trivial bundle to E with fiber F" is the locally trivial fibration $$E\times_{G}F:=(E\times F)/G\rightarrow B$$ $$[y,f]\mapsto p(y)$$ with local trivialization induced by those for E. My question is: is the structure group of this new bundle the same of E? I would say that, since the associated bundle has fiber F its structure group should be just Diff(F) in the smooth category.In particular, if F=H<G is a subgroup of the Lie group G, structure group is H itself. Is it right? If not, can someone explain why?

Another question: if we start from a vector bundle $$p:E\rightarrow B$$ then we know that its structure group is $$GL(n,\mathbb{R})$$ and it's usually stated that one can build a principal bundle defining $$E_{GL}:=\bigcup U_{\alpha}\times GL(n,\mathbb{R})/\sim$$ where $$\sim$$ is the equivalence relation defined as $$(b,f)\sim(\widetilde{b},\widetilde{f})\iff(\widetilde{b}=b,\widetilde{f}=f\phi_{\alpha\beta}(b))\forall b\in U_{\alpha}\cap U_{\beta}$$, $$\phi_{\alpha\beta}$$ denoting the clutching functions of the initial vector bundle. My question is:- How can we be sure that this is a principal bundle? By definition the local trivializations should be equivariant etc.. and to me this is not obvious with that construction.

• The structure group is a structure on a bundle, not a property of the bundle. May 3 at 9:22
• So $E\times_{G}F$ has no structure group a priori? Could you be more explicit? May 3 at 9:34

Yes, if $$p:E \rightarrow B$$ is a principal $$G$$-bundle and $$G$$ acts on $$F$$, the associated bundle $$E \times_G F$$ has same transition functions of $$E$$ (this follows from the fact that the local trivializations are induced by those of $$E$$), fiber $$F$$ and structure group $$G$$ (hence, the same of the principal $$G$$-bundle).
In particular, when we are talking about the reduction of the structure group to $$H < G$$, the bundle that we construct as $$E' \times_H G$$ has fiber $$G$$ and structure group $$H$$ (where we denote by $$E'$$ the principal $$H$$-bundle constructed via the transition functions with values in $$H$$).