Definition:Faithful Group Action
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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
$\phi$ is faithful if and only if $e$ is the only element if $G$ which acts trivially:
- $\forall g \in G: \paren {\forall x \in X: g * x = x \implies g = e}$
Definition 2
$\phi$ is faithful if and only if its permutation representation is injective.
Also known as
A faithful group action is also known as an effective group action.