What are the group axioms? First, you need an identity element: . By cancellation, we find . Next, check that associativity holds:
This follows from associativity and commutativity of real numbers.
Next, check that each element has an inverse. Let . You want to find so that . So, . Solving for we find . Hence, exists so long as .
So, no, it is not a group, but is a group.