These groups are identified respectively as the gauge transformations that become constant asymptotically and those that become the identity asymptotically. In the Abelian case G=U(1) the quotient is then identified as the group of global gauge symmetries, i.e. U(1) itself. However, known derivations of this claim are imprecise, both mathematically and conceptually. Based on work that came out of my master thesis (https://arxiv.org/abs/2502.16151 and https://arxiv.org/abs/2504.17483), I derive the physical gauge group rigorously for both Abelian and non-Abelian gauge theory. The main new point is that the requirement to restrict to boundary-preserving transformations does not follow from finiteness of energy only, but also from the requirement that the Lagrangian of Yang-Mills theory be defined on a tangent bundle to configuration space. I then explain why the quotient consists precisely of a copy of the global gauge group for every homotopy class of gauge transformations. If time permits I consider the addition of a Higgs field as a reference frame and show that asymptotic boundary conditions differ in the unbroken and broken phases. This can be extended to axiomatic QFT, where it can be shown that the Higgs mechanism is an instance of global gauge symmetry breaking.