[[Group action]]
# Free group action

A [[group action]] of $G$ on $\Omega$ is called **free** or **semiregular** iff the [[Stabilizer group]] $G_{\omega}$ of every $\omega \in \Omega$ is $\{ 1 \}$, #m/def/group i.e. $g\omega \neq \omega$ for all $g \neq 1$.

## Properties

1. A free group action is necessarily [[Effective group action|effective]]. ^P1

> [!check]- Proof of 1
> Since $g\omega \neq \omega$ for all $g \neq 1$ and $\omega \in \Omega$,
> the induced automorphism $\alpha_{\omega} \in \Aut(\Omega)$ cannot be identity for such a $g$,
> hence $\ker \alpha_{-}$ is a [[group monomorphism]]
> proving [[#^P1]].
> <span class="QED"/>

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#state/tidy | #lang/en | #SemBr