Q:

Is there a 1-1 correspondence between symmetry and group theory?

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?

A:

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.

