Faithful action

Definition

Let a group $G$ act on a set $X$. The action is faithful when, for each $g \in G$, $g \cdot x = x$ for all $x \in X$ implies $g = e$.

Remarks

The action is faithful if and only if the associated homomorphism $G \to \operatorname{Sym}(X)$ is injective. So a faithful action identifies $G$ with a subgroup of $\operatorname{Sym}(X)$. The action of $G$ on itself by conjugation is faithful if and only if the centre of $G$ is trivial.

Explanations

L1 human

In a faithful action, each element of the group that is not the identity moves at least one point. So you can tell the group elements apart by watching how they move the points. The group is then a group of permutations of the set.