\begin{definition}[Definition:Permutation Representation/Group Action]
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 $\phi_g$.
\end{definition}