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 $$\phi_g \left({x}\right) = g * x$$, and its inverse is $$\phi_{g^{-1}}: X \to X$$.

These bijections are sometimes called transformations of $$X$$.

In this context, the group doing the action is referred to as a group of transformations, or a transformation group.

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.