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 P → → X with a structure Lie group G , a gauge group is defined to be a group of its vertical automorphisms. This group is isomorphic to the group G ( X ) of global sections of the associated group bundle P ~ ~ → → X whose typical fiber is a group G which acts on itself by the adjoint representation. The unit element of G ( X ) is a constant unit-valued section g ( x ) = 1 of P ~ ~ → → X .