Definition:Group Action Induced on Subgroup

From ProofWiki
Jump to navigation Jump to search



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