nLab orbit-stabilizer theorem

Idea

The orbit-stabilizer theorem says that a transitive action of a group $G$ on some $X$ is equivalent to the canonical action of $G$ on the cosets $G/Stab_G$ by the stabilizer subgroup $G_x \coloneqq Stab_g(x)$ of any element $x \in X$, identifying $X$ with the $G$-orbit of $x$;

$$X \simeq Orbit_G(x) \simeq G/Stab_G(x) \mathrlap{\,.}$$

For smooth actions

In differential geometry:

Proposition

For

• $G$ a Lie group,

• $G \curvearrowright X$ a homogeneous smooth $G$-manifold, hence with a transitive smooth $G$-action,

• $x \colon \ast \to X$ a point,

then

1. the stabilizer group of $x$ under $G$ is a closed subgroup $G_x \subset G$,

2. the map $G/G_x \longrightarrow X \;\colon\; g \cdot G_x \mapsto g \cdot x$ is an equivariant diffeomorphism.

(cf. Warner 1983 Thm. 3.62, Lee 2012, Thm. 21.18)

References

For finite groups acting on sets:

• The Orbit-Stabilizer Theorem [pdf]

See also:

• Wikipedia: Orbit-stabilizer theorem

For Lie group actions on smooth manifolds:

• Frank W. Warner, thm. 3.62 in: Foundations of Differentiable Manifolds and Lie Groups, Graduate Texts in Mathematics 94 (1983) [doi:10.1007/978-1-4757-1799-0]

• John Lee, Thm. 21.18 in: Introduction to Smooth Manifolds, Springer (2012) [doi:10.1007/978-1-4419-9982-5, book webpage, pdf]