\begin{definition}[Definition:Faithful Group Action/Definition 1]
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''' {{iff}} $e$ is the only element if $G$ which acts trivially:
:$\forall g \in G: \paren {\forall x \in X: g * x = x \implies g = e}$
\end{definition}