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 $X$.

Also see

• Definition:Group Action Induced on Stable Subset