Reduction of the structure group

Reduction of the structure group

One can ask if a principal ${\displaystyle G}$-bundle over a group${\displaystyle G}$ "comes from" a subgroup${\displaystyle H}$ of ${\displaystyle G}$. This is called reduction of the structure group (to ${\displaystyle H}$), and makes sense for any map ${\displaystyle H\to G}$, which need not be an inclusion map (despite the terminology).