= Faithful group action
{title2=$\ker\rho=\{e\}$}
{wiki}
= Faithful
{synonym}
A group action is faithful when only the identity acts as the identity on every point.
For a <group action> of $G$ on a <set> $X$, form the <group homomorphism>
$$
\rho:G\longrightarrow\operatorname{Sym}(X),\qquad \rho(g)(x)=g\cdot x.
$$
A <faithful group action> means that $\rho$ is <injective>: different <group> elements induce different <permutations> of $X$. Equivalently, its <group kernel> is trivial:
$$
K=\ker\rho=\{g\in G:g\cdot x=x\text{ for every }x\in X\}=\bigcap_{x\in X}G_x=\{e\}.
$$
Here $G_x$ is the <stabilizer subgroup> of $x$, and $e$ is the <identity element>.<Group Actions by Keith Conrad>[Theorem 1.7, printed page 3, and Definition 2.17, printed page 9: actions as homomorphisms and faithfulness]{ref}
The test is global. A <faithful group action> requires each nonidentity <group> element to move at least one point; a <free group action> requires it to move every point. In the <faithful coset action of the symmetric group on three letters>, $(12)$ fixes one <coset> and exchanges the other two. Thus a nontrivial <stabilizer subgroup> does not rule out faithfulness. The distinction follows directly from the order of the quantifiers:
$$
\text{faithful: }\forall g\ne e\ \exists x:\ g\cdot x\ne x,\qquad
\text{free: }\forall g\ne e\ \forall x:\ g\cdot x\ne x.
$$
On a nonempty <set>, a <free group action> is therefore a <faithful group action>. A <transitive group action> instead asks whether every point can reach every other; it allows several <group> elements to perform exactly the same rearrangement. The <nonfaithful coset action of the symmetric group on three letters> is a counterexample to the claim that every <transitive group action> is a <faithful group action>.
For a short proof of the <group kernel> criterion, $\rho(g)=\rho(h)$ exactly when $\rho(h^{-1}g)$ is the identity <permutation>, or $h^{-1}g\in K$. If $K=\{e\}$, this forces $g=h$; if $K$ contains a nonidentity element, that element and $e$ act identically. Thus a <faithful group action> turns $G$ into an actual <subgroup> of a <symmetric group>, retaining its <group operation> as <function composition>. Choosing a suitable <coset action> can give a much smaller <set> of labels than the <regular group action> used in <Cayley's theorem>.
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