Correspondence Between Group Actions and Permutation Representations

Theorem
Let $G$ be a group.

Let $X$ be a set.

There is a one-to-one correspondence between group actions of $G$ on $X$ and permutation representations of $G$ in $X$, as follows:

Let $\phi : G \times X \to X$ be a group action.

Let $\rho : G \to \operatorname{Sym}(x)$ be a permutation representation.

The following are equivalent:


 * $(1): \quad$ $\rho$ is the permutation representation associated to $\phi$


 * $(2): \quad$ $\phi$ is the group action associated to $\rho$