Free Action - Knowlpedia

A group action of a group $G$ on a set $X$ is free if every stabilizer is trivial:

G_x=\{e\}\qquad(x\in X).

Equivalently, if $g\cdot x=x$ for some $x\in X$ , then $g=e$ .

Remarks

Free actions are one half of the definition of a regular action (free + transitive).

Examples

The left translation action of $G$ on itself is free.
The action of $G$ on the coset space $G/H$ by left multiplication is free iff $H=\{e\}$ .
The action of $C_n$ on the vertices of a regular $n$ -gon by rotation is free when restricted to the vertex set (no nontrivial rotation fixes a vertex).