Definition:Faithful 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:

$\forall \tuple {g, x} \in G \times X: g * x := \map \phi {g, x} \in X$

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.

Also see

• Equivalence of Definitions of Faithful Group Action
• Definition:Kernel of Group Action
• Definition:Free Group Action

• Results about faithful group actions can be found here.