Stabilizer

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

The stabilizer $G_x$ is always a subgroup of $G$. 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 $|G\cdot x| = [G:G_x]$.

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 , $\{g\in G:gx=xg\}$.