Faithful Action

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.

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).