Definition:Permutation Representation/Group Action

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

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

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

Also see

 * Group Action determines Bijection, which shows that $\phi_g \in \operatorname{Sym} \left({X}\right)$


 * Group Action Defines Permutation Representation, which shows that this defines a homomorphism