Faithful group action

Definition in action terms
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 $$g.s \ne s$$.

Definition in terms of homomorphisms
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.