# Definition:Kernel of Group Action

## Definition

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

Let $X$ be a set.

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

### Definition 1

The kernel of the group action is the set:

$G_0 = \left\{{g \in G: \forall x \in X: g \cdot x = x}\right\}$

### Definition 2

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