QED from Abelian $U(1)_Q$ group In all quantum field theory books, you read something like

QED is a QFT from Abelian $U(1)_Q$ that describes the electromagnetic interaction ...

But I fail to understand what is really meant by that. How is a QFT linked to group theory? How do you come up with this conclusion? If someone can help me make the link, that would be fantastic!
 A: The basic reasoning is presented in Weinberg's "The Quantum Theory of Fields" Volume 1. It goes along the following lines: we want to consider the possible quantum mechanical theories of relativistic particles. These are, by definition, constrained by Poincaré symmetry. Moreover, we want to focus on theories that obey the so-called "cluster decomposition principle", which basically asserts that far separated experiments must yield uncorrelated results.
In that regard, what Weinberg exposes in Chapters 2 - 5 is that the simplest way of constructing such theories is by taking the creation and annihilation operators $a^\dagger(\mathbf{p},\sigma)$, $a(\mathbf{p},\sigma)$ and embedding these into fields that transform in representations of the Lorentz group. One then uses these fields to construct interaction terms.
That said, what Weinberg exposes is that if you take the photon creation and annihilation operators and try to embed those into a vector field $A_\mu(x)$ you get one object that does not transform properly. More precisely, under Lorentz transformations it behaves like $$U(\Lambda)A^\mu(x)U^{-1}(\Lambda)=\Lambda^\mu_{\phantom\mu\nu}A^\nu(\Lambda x)+\partial^\mu \Omega(x;\Lambda).\tag{1}$$
In other words: a vector field which encodes the photon transforms as a vector field only up to a gauge transformation. Such an object can still be used to construct a valid interaction provided one demands gauge invariance, so that the inhomogeneous shift that appears in (1) in the end of the day does not matter. One then may show that the possible interactions must be of the form $A_\mu j^\mu$ where $j^\mu$ is a conserved current $\partial_\mu j^\mu =0$.
In turn after all of this, we may look to the theory of such a vector field $A_\mu$ and a fermion field $\psi$ and observe that what actually happend is that the free $\psi$ theory had one global $U(1)$ symmetry and the inclusion of $A_\mu$ promoted it to a local symmetry. One may then try to generalize this: what if we have a theory with global symmetry characterized by some Lie group $G$ and we try to make it local? It turns out that one has to introduce one analogous $A_\mu$ which is now seem to be $\mathfrak{g}$-valued where $\mathfrak{g}$ is the Lie algebra of $G$. In this way one lands on non-abelian gauge theories.
By the way, if one has studied the theory of principal bundles and connections on principal bundles, one recognizes (1) as the transformation law for the local representative of a connection on a $U(1)$-bundle upon a change of trivialization of the bundle. This is another, geometric, route to the generalization I have mentioned that builds upon the true geometric nature of the gauge field: it is actually a connection on a principal $G$-bundle!
In summary, that is the way in which we get the kinds of things you mention in your question. To describe the photon through a vector potential gauge invariance is in a sense forced upon us, and a natural generalization leads to a class of theories, namely the gauge theories, which are naturally characterized by Lie groups $G$.
A: The answer by @Gold is perfect, I would just add the following supplementary point:
Notice that the $U(1)$ gauge redundancy is already present in classical electrodynamics when you formulate it using electromagnetic potentials. However, in classical mechanics, you can construct the full theory of electrodynamics without ever referring to potentials if you want to do so. On the other hand, in quantum mechanics, there isn't a satisfactory way (i.e., a way that preserves unitarity, Lorentz invariance, and locality) to formulate the electromagnetic theory using only the fields even if only the fields correspond to physical degrees of freedom. So, we necessarily formulate the theory in terms of electromagnetic potential (i.e., degrees of freedom that are $U(1)$ fold redundant in some loose sense), and thus, you need to make sure you are keeping in mind that your theory has this redundancy that you need to take care of while making predictions, i.e., you need to identify the potential configurations that are related by $U(1)$ transformations as physically identical. See, for example, the Faddeev-Popov path integral procedure where this identification process is quite manifest.
So, the simple point is that you can avoid the gauge structure of field theories in classical mechanics but you can't do so in quantum mechanics. And the gauge structure of field theories has the structure of Lie groups. And that is the reason groups are omnipresent in quantum field theory. In some sense, it is literally the geometry of how we are organizing the copies of the physical degrees of freedom in our theory -- and so it very much defines the structure of the theory. However, notice that this is about how the redundancy is organized -- and dualities have shown us that there can be two different gauge structures in which we can organize the same physical degrees of freedom, i.e., a quantum field theory that describes the same physical degrees of freedom might be formulated both as a $G_1$ gauge theory and a $G_2$ gauge theory where $G_{1,2}$ are some Lie groups.
