Stabilizer of Group Action

up:: Group Action

Given a Group Action of a Group $G$ upon a set $X$, we have that the stabilizer of a point $x\in X$ is the set of group elements which keep it intact.

$$Sta{b}_x=\{g\in G\mid g\cdot x=x\}\subseteq G$$

Properties

Note that Stabilizers are a subgroup.

In fact, we have that a group action’s stabilizers are only trivial iff the group action is a Free Group Action.