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 $G$ over some set $X$.

Let $x\in X$, and $Or{b}_x$ its Orbit.

Construct a map

$$\begin{array}{rl}\phi :G& \to Or{b}_x\\ & g\mapsto g\cdot x\end{array}$$

Note that Group actions induce a surjection between the group and a point’s orbit, since all points $g\cdot x\in Or{b}_x$ have at least some $g\in G$ from which they get from $x$ to $g\cdot x$.

If the action is free, then $\phi$ must be injective. Let $g\cdot x=h\cdot x\in Or{b}_x$. Then $x=(g^{-1}h)\cdot x$, for which $g^{-1}h=e_G⟺g=h$ (due to the action being free).

Thus, free group actions induce a bijection between $G$ and a point’s orbit $Or{b}_x$.