# Faithful group action

A group action of a group $G$ on a set $S$ is termed faithful or effective if for any non-identity elemnet $g \in G$, there is $s \in S$ such that $g.s \ne s$.
A group action of a group $G$ on a set $S$ is termed faithful or effective if the corresponding homomorphism from $G$ to $\operatorname{Sym}(S)$ is an injective homomorphism.