Group_Action_defines_Permutation_Representation

Let $\map \Gamma X$ be the set of permutations on a set $X$. Let $G$ be a group. Let $\phi: G \times X \to X$ be a group action. For $g \in G$, let $\phi_g: X \to X$ be the mapping defined as $\map {\phi_g} x = \map \phi {g, x}$ Let $\tilde \phi: G \to \map \Gamma X$ be the mapping associated to $\phi$, defined by $\map {\tilde \phi} g := \phi_g$ Then $\tilde \phi$ is a group homomorphism.