Group Action determines Bijection

Theorem
Let $*$ be a group action of $G$ on $X$.

Then each $g \in G$ determines a bijection $\phi_g: X \to X$ given by:
 * $\map {\phi_g} x = g * x$

Its inverse is:
 * $\phi_{g^{-1} }: X \to X$.

These bijection are sometimes called transformations of $X$.

Proof of Injectivity
Let $x, y \in X$

Then:

Thus $\phi_g$ is an injection.

Proof of Surjectivity
Let $x \in X$.

Then:

Thus a group action is a surjection.

So a group action is an injection and a surjection and therefore a bijection.

Also see

 * Definition:Permutation Representation