Faithful Action - Imported Knowlpedia Content

A group action of a group $G$ on a set $X$ is faithful if its kernel is the trivial subgroup. Equivalently, the associated homomorphism $G\to \mathrm{Sym}(X)$ is a monomorphism, i.e. injective.

Remarks

Faithful actions are exactly those that realize $G$ as a subgroup of a permutation group; this viewpoint underlies Cayley's theorem.

Examples

The left translation action of $G$ on itself is faithful (and in fact regular).
The conjugation action of $G$ on itself is faithful iff $Z(G)=\{e\}$ .
If $G$ is abelian and nontrivial, the conjugation action is not faithful (every element acts trivially).