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.

Also see

 * Definition:Permutation Representation