---- > [!definition] Definition. ([[faithful group action]]) > Let $G$ be a [[group]] and $X$ be set. A [[group action]] $\begin{align} G \times X & \to X \\ (g,x) \mapsto g \cdot x \end{align}$ is said to be **faithful** if the only element $g$ for which $g \cdot x=x$ *for all* $x \in X$ is the identity $e_{G}$, that is, if the corresponding [[covariant functor|functor]] $G \to \text{Aut}(X)$ is a [[faithful functor]]. ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```