Free group action

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

Properties

A free group action is necessarily effective.

Proof of 1

Since $𝑔 𝜔 \neq 𝜔$ for all $𝑔 \neq 1$ and $𝜔 \in Ω$ , the induced automorphism $𝛼 𝜔 \in A u t (Ω)$ cannot be identity for such a $𝑔$ , hence $k e r 𝛼 -$ is a group monomorphism proving ^P1.

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