\begin{definition}[Definition:Group Action/Permutation Representation]
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 $\map {\rho_g} x = \map \phi {g, x}$.
\end{definition}