Orbit–Stabilizer Theorem

For a group action, an orbit is in bijection with a coset space G/Stab(x)

Orbit–Stabilizer Theorem

Orbit–Stabilizer Theorem. Let $G$ be a group acting on a set $X$ via a group action . For $x \in X$ , let $\operatorname{Orb}(x)$ be the orbit of $x$ and let $\operatorname{Stab}(x)$ be the stabilizer of $x$ . Then the map

G/\operatorname{Stab}(x) \to \operatorname{Orb}(x), \qquad g\,\operatorname{Stab}(x) \mapsto g\cdot x,

is a bijection. In particular, if $G$ is finite then

|\operatorname{Orb}(x)| = [G:\operatorname{Stab}(x)] \quad \text{and} \quad |G| = |\operatorname{Orb}(x)|\cdot |\operatorname{Stab}(x)|.

This theorem converts problems about orbits into problems about cosets and index . It is the main input for the class equation and many counting arguments.

Proof sketch. The map $g \mapsto g\cdot x$ is constant on left cosets of $\operatorname{Stab}(x)$ and surjects onto $\operatorname{Orb}(x)$ . Two elements $g,h$ give the same point iff $h^{-1}g \in \operatorname{Stab}(x)$ , which is exactly the equivalence relation defining the coset space.