Free Group Actions induce a bijection between the group and a point's orbit

up:: Free Group Action

Let there be a Free Group Action of a Group over some set .

Let , and its Orbit.

Construct a map

Note that Group actions induce a surjection between the group and a point’s orbit, since all points have at least some from which they get from to .

If the action is free, then must be injective. Let . Then , for which (due to the action being free).

Thus, free group actions induce a bijection between and a point’s orbit .