Definition:Group Action Induced on Subgroup

Definition
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
 * $\rho \circ \iota : H \to \operatorname {Sym} \left({X}\right)$

where:


 * $\iota : H \to G$ is the inclusion homomorphism
 * $\rho$ is the permutation representation of $\phi$

Also see

 * Definition:Group Action Induced on Stable Subset