= Structure group of a vector bundle
{title2=$g_{ij}:U_i\cap U_j\longrightarrow G\subseteq\operatorname{GL}(k,\mathbb C)$}
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>.
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