simply transitive
Let be a group acting on a set . The action is said to be simply transitive if it is transitive and there is a unique such that .
Theorem.
A group action is simply transitive if and only if it is free and transitive
Proof.
Necessity follows since implies that because also. Now assume the action is free and transitive and we have elements and such that and . Then hence because the action is free. Thus and so the action is simply transitive.∎