TITLE: How to prove closure property for $x\cdot y = \frac{x + y}{1 + \frac{xy}{c^2}}$?
QUESTION [5 upvotes]: Given $G$ has elements in the interval $(-c, c)$. Group operation is defined as:
$$x\cdot y = \frac{x + y}{1 + \frac{xy}{c^2}}$$
How to prove closure property to prove that G is a group?

REPLY [4 votes]: Another approach is to write $x=c\tanh a,\,y=c\tanh b$ with $a,\,b\in\Bbb R$ (these are called rapidities), then show $x\cdot y=c\tanh (a+b)$. This proves $G$ is isomorphic to $(\Bbb R,\,+)$.