Definition:Permutation Representation

Definition
Let $X$ be a set.

Let $G$ be a group.

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

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

Define for $g\in G$ the mapping $\phi_g : X \to X$ by:
 * $\phi_g (x) = \phi(g, x)$.

By Group Action determines Bijection, $\phi_g\in\operatorname{Sym}(X)$ for all $g\in G$.

The permutation representation of $G$ associated the group action is the group homomorphism $G\to\operatorname{Sym}(X)$ which sends $g$ to $\phi_g$.

Also see

 * Permutation Representation is Group Homomorphism