The professor in my class of mathematical physics introduces the definition of groups and said that group theory is the mathematics of symmetry.
He gave also some examples of groups such as the set of all real numbers under addition.
My question is what symmetry are we talking about here when we consider the set of all real numbers under addition?
On one hand, a group $G$ is a purely mathematical concept.
On the other hand, given a system $S$ on which a group $G$ acts $G\times S \to S$. The system $S$ then constitutes a (possibly non-linearly realized) representation of the group $G$. The system $S$ can in certain cases be invariant under the group action, and in such cases we say the system $S$ possesses a $G$-symmetry.
In physics, $S$ is typical a set of equations, or an action (not to be confused with the notion of a group action).
The Abelian group $G=(\mathbb{R},+)$ that OP mentions is implemented in many physical systems. It could be translation symmetry of a physical system in a given direction, as Maksim Zholudev writes in a comment. But this is just one out of many possibilities, and there is not a 1-1 correspondence between groups and symmetries in a strict mathematical sense.
Finally, note that the notion of a group $G$ may be weakened/generalized in several ways, e.g. to a groupoid.
There is a one to one correspondence between symmetry and group theory for the simple reason that if A is a symmetry and B is a symmetry, then so is B followed by A. This implies that symmetries form a group, where the group law is composition of maps (a symmetry is a map).
Translation along the real axis is the physical symmetry of time translation. As pointed out by Wiener, it is an important fact that if we begin an experiment at time $ t =0$ and measured the results at time $t = 4$, the answers will be, physically, the same as if we had started the experiment at time $t=21$ and measured the results at time $t=25$.
This is the reason why Fourier analysis is useful... whenever you have a symmetry group, the representations of that group are useful. The representations of the translation group are the exponential functions so Fourier analysis, the decomposition of an arbitrary function into combinations of different exponential functions, is useful.
