Stabilizer - Knowlpedia

Let a group action of a group $G$ on a set $X$ be given. For $x\in X$ , the stabilizer of $x$ is

G_x := \{g\in G : g\cdot x = x\}.

The stabilizer $G_x$ is always a subgroup of $G$ .

Examples

For the natural action of $S_n$ on $\{1,\dots,n\}$ , the stabilizer of $1$ is the subgroup of permutations fixing $1$ , which is isomorphic to $S_{n-1}$ .
For the left translation action of $G$ on itself, the stabilizer of any $x\in G$ is trivial.
For the conjugation action of $G$ on itself, the stabilizer of an element $x$ is its centralizer, $\{g\in G:gx=xg\}$ .

Remarks

Together with the orbit $G\cdot x$ , it controls the action: the orbit-stabilizer theorem gives a bijection between $G/G_x$ and $G\cdot x$ , and in the finite case implies $|G\cdot x| = [G:G_x]$ .