Free Action

A group action of a group $G$ on a set $X$ is free if for every $x\in X$, the stabilizer $G_x$ is the trivial subgroup . Equivalently, the condition “$g\cdot x = x$ for some $x$” forces $g=e$.

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).