Transitive Group Actions are Surjective

up:: Transitive Group Action

Let be the base space, and a Group. Let be a Transitive Group Action.

Let . We can know the inverse image of this group action, , as follows:
Since it is a transitive action, then . Acting on the left by yields that

(the two last equalities follow from identity and compatibility of a group action.)

Thus, a transitive group action is surjective.