Definition:Faithful Group Action

Definition
Let $G$ be a group.

Let $S$ be a set.

Let $*: G \times S \to S$ be a group action.

Then $*$ is faithful iff:
 * $\forall g \in G: g * s_1 = g * s_2 \implies s_1 = s_2$

That is, iff $*: G \times S \to S$ is an injection.

Also see

 * Definition:Free Group Action