stabilizer

Stabilizer under group action

Let $G$ be a group that acts on a set $X$. For $x\in X$, the stabilizer is $G_x=\{g\in G|gx=x\}$

The stabilizer is all group elements that stabilize $x$ in the sense that they don’t move it around under the group action.

Notes

• If the action is free then $G_x=\{e\}$.
• The stabilizer of any element $G_x$ is a subgroup of $G$.