# Definition:Group Action Induced on Subgroup

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## 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:

- $\rho \circ \iota : H \to \struct {\map \Gamma X, \circ}$

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 $X$.