Lemma 39.10.3. Situation as in Definition 39.10.2, The action $a$ is free if and only if

\[ G \times _ S X \to X \times _ S X, \quad (g, x) \mapsto (a(g, x), x) \]

is a monomorphism.

Lemma 39.10.3. Situation as in Definition 39.10.2, The action $a$ is free if and only if

\[ G \times _ S X \to X \times _ S X, \quad (g, x) \mapsto (a(g, x), x) \]

is a monomorphism.

**Proof.**
Immediate from the definitions.
$\square$

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