\begin{definition}[Definition:Group Action Induced on Subgroup]
Let $G$ be a group.
Let $X$ be a set.
Let $\phi : G \times X \to X$ be a group action.
Let $H \le G$ be a subgroup.
The '''group action induced on $H$''' is the restriction of $\phi$ to $H \times X$.
Equivalently, the '''group action induced on $H$''' is the group action associated to the permutation representation:
:$\rho \circ \iota : H \to \struct {\map \Gamma X, \circ}$
where:
:$\iota : H \to G$ is the inclusion homomorphism
:$\rho$ is the  permutation representation of $\phi$
:$\struct {\map \Gamma X, \circ}$ is the symmetric group on $X$.
\end{definition}