Definition:Group Action/Permutation Representation

Definition
Let $G$ be a group.

Let $X$ be a set. Let $\operatorname{Sym} \left({X}\right)$ be the group of permutations on $X$.

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

The group action of $G$ associated to the permutation representation $\rho$ is the group action $\phi : G \times X \to X$ defined by:
 * $\phi (g, x) = \rho(g)(x)$

Also see

 * Permutation Representation Defines Group Action, where it is shown that $\phi$ is indeed a group action