Regular group action

A group action is regular if it is transitive and free. This is equivalent to saying that given two points $x,y\in X$ there is exactly one $g\in G$ such that $gx=y$. In this case, $X$ is called a $G$-torsor or a principal homogeneous space. The action of $G$ on itself is regular. KEEP AN EYE: Peter Olver uses the concept of group acting regularly. I think it has to do with the concept of \textit{regular map}, \textit{regular submanifold} and \textit{proper action} (see just below). I have to think it yet.