Let $G$ be a group with identity element $e$, let $X$ be a set, and let $G$ act on $X$ by the action map
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\begin{align*}
G \times X \to X, \quad (g,x) \mapsto g \cdot x.
\end{align*}
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Let $\operatorname{Sym}(X)$ denote the group of bijections $X \to X$, and let
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\begin{align*}
\rho: G \to \operatorname{Sym}(X)
\end{align*}
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be the associated action homomorphism, defined by $\rho(g)(x)=g \cdot x$ for all $g \in G$ and $x \in X$. Then the action of $G$ on $X$ is faithful if and only if