Definition:Faithful Group Action/Definition 1

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.

$\phi$ is faithful $e$ is the only element which acts trivially:
 * $\forall g \in G : \left({\forall x \in X: g * x = x \implies g = e}\right)$

That is, $G$ is an effective transformation group if the kernel $G_0$ equals $\left\{{e}\right\}$.

Also known as
A faithful group action is also known as an effective group action.

Also see

 * Definition:Effective Transformation Group