Regular group action

A group $𝐺$ is said to act regularly or sharply transitively on $𝑀$ if the action is both free and transitive, group i.e. all point stabilizers are ${𝑒}$ and each orbit $𝐺 𝑚 = 𝑀$ covers the whole space. Equivalently, there exists exactly one $𝑔 \in 𝐺$ such that $𝑔 𝑚 = 𝑚'$ for all $𝑚, 𝑚' \in 𝑀$ .