Definition:Kernel of Group Action

This page is about Kernel of Group Action. For other uses, see Kernel.

Definition

Let $G$ be a group with identity $e$.

Let $X$ be a set.

Let $* : G\times X\to X$ be a group action.

Definition 1

The kernel of the group action is the set:

$G_0 = \set {g \in G: \forall x \in X: g * x = x}$

Definition 2

The kernel of the group action is the kernel of its permutation representation.

Also see

• Equivalence of Definitions of Kernel of Group Action
• Kernel of Group Action is Normal Subgroup
• Definition:Faithful Group Action