# Orbit-stabiliser theorem

Let $$G$$ be a finite group, act­ing on a set $$X$$. Let $$x \in X$$. Writ­ing $$\mathrm{Stab}_G(x)$$ for the sta­bil­iser of $$x$$, and $$\mathrm{Orb}_G(x)$$ for the or­bit of $$x$$, we have

$$|G| = |\mathrm{Stab}_G(x)| \times |\mathrm{Orb}_G(x)|$$
where $$| \cdot |$$ refers to the size of a set.

This state­ment gen­er­al­ises to in­finite groups, where the same proof goes through to show that there is a bi­jec­tion be­tween the left cosets of the group $$\mathrm{Stab}_G(x)$$ and the or­bit $$\mathrm{Orb}_G(x)$$.

# Proof

Re­call that the sta­bil­iser is a sub­group of the par­ent group.

Firstly, it is enough to show that there is a bi­jec­tion be­tween the left cosets of the sta­bil­iser, and the or­bit. In­deed, then

$$|\mathrm{Orb}_G(x)| |\mathrm{Stab}_G(x)| = |\{ \text{left cosets of} \ \mathrm{Stab}_G(x) \}| |\mathrm{Stab}_G(x)|$$
but the right-hand side is sim­ply $$|G|$$ be­cause an el­e­ment of $$G$$ is speci­fied ex­actly by spec­i­fy­ing an el­e­ment of the sta­bil­iser and a coset. (This fol­lows be­cause the cosets par­ti­tion the group.)

## Find­ing the bijection

Define $$\theta: \mathrm{Orb}_G(x) \to \{ \text{left cosets of} \ \mathrm{Stab}_G(x) \}$$, by

$$g(x) \mapsto g \mathrm{Stab}_G(x)$$

This map is well-defined: note that any el­e­ment of $$\mathrm{Orb}_G(x)$$ is given by $$g(x)$$ for some $$g \in G$$, so we need to show that if $$g(x) = h(x)$$, then $$g \mathrm{Stab}_G(x) = h \mathrm{Stab}_G(x)$$. This fol­lows: $$h^{-1}g(x) = x$$ so $$h^{-1}g \in \mathrm{Stab}_G(x)$$.

The map is in­jec­tive: if $$g \mathrm{Stab}_G(x) = h \mathrm{Stab}_G(x)$$ then we need $$g(x)=h(x)$$. But this is true: $$h^{-1} g \in \mathrm{Stab}_G(x)$$ and so $$h^{-1}g(x) = x$$, from which $$g(x) = h(x)$$.

The map is sur­jec­tive: let $$g \mathrm{Stab}_G(x)$$ be a left coset. Then $$g(x) \in \mathrm{Orb}_G(x)$$ by defi­ni­tion of the or­bit, so $$g(x)$$ gets taken to $$g \mathrm{Stab}_G(x)$$ as re­quired.

Hence $$\theta$$ is a well-defined bi­jec­tion.

Children:

Parents:

• Group action

“Groups, as men, will be known by their ac­tions.”