Gauge group in the context of Gauge gravitation theory

A gauge group is a group of gauge symmetries of the Yang–Mills gauge theory of principal connections on a principal bundle. Given a principal bundle ${\displaystyle P\to X}$ with a structure Lie group ${\displaystyle G}$, a gauge group is defined to be a group of its vertical automorphisms, that is, its group of bundle automorphisms. This group is isomorphic to the group ${\displaystyle G(X)}$ of global sections of the associated group bundle ${\displaystyle {\widetilde {P}}\to X}$ whose typical fiber is a group ${\displaystyle G}$ which acts on itself by the adjoint representation. The unit element of ${\displaystyle G(X)}$ is a constant unit-valued section ${\displaystyle g(x)=1}$ of ${\displaystyle {\widetilde {P}}\to X}$.

At the same time, gauge gravitation theory exemplifies field theory on a principal frame bundle whose gauge symmetries are general covariant transformations which are not elements of a gauge group.