Definition:Permutation Representation/Group Action

Definition
Let $G$ be a group.

Let $X$ be a set.

Let $\struct {\map \Gamma X, \circ}$ be the symmetric group 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:
 * $\map {\phi_g} x = \map \phi {g, x}$

The permutation representation of $G$ associated to the group action is the group homomorphism $G \to \struct {\map \Gamma X, \circ}$ which sends $g$ to $\phi_g$.

Also see

 * Group Action determines Bijection, which shows that $\phi_g \in \struct {\map \Gamma X, \circ}$


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