Group actions induce a surjection between the group and a point's orbit

up:: Group Action

Let $G$ be a Group acting upon a set $X$. Let $x\in X$ and $Or{b}_x$ its Orbit.

Then the map

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

is a surjection, since all points in the orbit have at least some $g\in G$ which take $x$ to it.