Stabilizer (isotropy subgroup)

Let $G$ act on a manifold $M$ via a smooth action .

For $x\in M$, the stabilizer (or isotropy subgroup) of $x$ is

It is a subgroup of $G$. If $G$ is a Lie group and the action is smooth, then $G_x$ is a closed subgroup of $G$, hence a Lie subgroup.

The stabilizer controls the orbit through $x$: the orbit map $G\to G\cdot x$ has kernel $G_x$, and (under mild hypotheses) the orbit is modeled on the homogeneous space $G/G_x$.

Examples

1. Rotations of the plane. For $SO(2)\curvearrowright \mathbb{R}^2$, the stabilizer of $0$ is all of $SO(2)$, while the stabilizer of any nonzero vector is trivial.
2. Coset actions. For the left action of $G$ on $G/H$ by $g\cdot (g'H)= (gg')H$, the stabilizer of the basepoint $eH$ is exactly $H$.
3. Conjugation. For the conjugation action of $G$ on itself, the stabilizer of an element $h$ is its centralizer $C_G(h)=\{g\in G\mid gh=hg\}$.