Definition:Group Action/Permutation Representation

Definition
Let $G$ be a group.

Let $X$ be a set. Let $\struct {\map \Gamma X, \circ}$ be the symmetric group on $X$.

Let $\rho: G \to \struct {\map \Gamma X, \circ}$ 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:
 * $\map \phi {g, x} = \map {\rho_g} x$

where $\rho_g : X \to X$ is the permutation representation associated to $\rho$ for $g \in G$ by $\map {\rho_g} x = \map \phi {g, x}$.

Also see

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