\begin{definition}[Definition:Kernel of Group Action/Definition 1]
Let $G$ be a group with identity $e$.
Let $X$ be a set.
Let $* : G\times X\to X$ be a group action.
The '''kernel of the group action''' is the set:
:$G_0 = \set {g \in G: \forall x \in X: g * x = x}$
\end{definition}