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 $\rho$

Proof
For $g\in G$, define the mapping $\phi_g : X \to X$ as:
 * $\phi_g(x) = \phi(g, x)$

Then $\rho$ is the permutation representation associated to $\phi$ :
 * $\forall g\in G : \rho(g) = \phi_g$

By Equality of Mappings, this is equivalent to:
 * $\forall g\in G : \forall x\in X : \rho(g)(x) = \phi_g(x)$

$\phi$ is the group action associated to $\rho$ :
 * $\forall g\in G : \forall x\in X : \phi(g, x) = \rho(g)(x)$.

Because $\phi_g(x) = \phi(g, x)$, they are equivalent.