Structure group of a vector bundle ($g_{ij}:U_i\cap U_j⟶G\subseteq \mathrm{GL}(k,\mathbb{C})$)

A complex vector bundle has structure group $G$ when it admits fiber-linear local trivializations whose transition matrices take values in $G$. Replacing frames by $h_i$ changes transitions to $h_i^{-1}{g}_{ij}{h}_j$ in the frame convention. Reducing the structure group means choosing new trivializations with values in a specified subgroup. The same transitions glue $U_i\times G$ by left multiplication to a right principal bundle.