Orbit-Stabiliser theorem

Orbit-Stabiliser Theorem

Let $G$ be a group acting on a set $X$ via $\lambda$. For all $x\in X$,

$$\boxed{|Gx|=[G:{\mathrm{Stab}}_G(x)].}$$

If $G$ is finite,

$$|Gx|=\frac{|G|}{|{\mathrm{Stab}}_G(x)|}.$$

The size of the Orbit of $x$ equals the index of its Stabiliser.

Proof Sketch

Let $S={\mathrm{Stab}}_G(x)$. Define a map $G/S\to Gx$ by $gS\mapsto \lambda (g)x$. This is

• Well-defined: $g_{1}S=g_{2}S⟺g_1^{-1}{g}_2\in S⟺\lambda (g_1^{-1}{g}_2)x=x⟺\lambda (g_1)x=\lambda (g_2)x$.
• Surjective: by definition of $Gx$.
• Injective: by the same equivalence.

So it’s a bijection. Hence $|Gx|=|G/S|=[G:S]$. $■$

Applications

1. Compute orbit sizes by computing stabilisers.
2. Rule out transitive actions: if $|X|\nmid |G|$, no transitive action of $G$ on $X$ exists. (Corollary: $D_{10}$ cannot act transitively on a triangle.)
3. Class equation in conjugation actions: $|G|=|Z(G)|+\sum |G|/|C_G(g_i)|$, summing over non-central conjugacy classes.
4. Sylow theorems: the proofs of Sylow’s theorems all use Orbit-Stabiliser at key steps.

Example: $S_4$ on $\{1,2,3,4\}$

• Action transitive, so $|G\cdot 1|=4$.
• ${\mathrm{Stab}}_{S_4}(1)\cong S_3$, $|\cdot |=6$.
• Check: $4=24/6$. ✓

Example: $S_n$ on Pairs

$S_n$ acts on $\{1,\dots ,n\}\times \{1,\dots ,n\}$ via $\lambda (g)(i,j)=(gi,gj)$. The orbit of $(1,1)$ is the diagonal $\{(i,i):i\in [n]\}$, so $|G\cdot (1,1)|=n$. The stabiliser is $\mathrm{Sym}(\{2,\dots ,n\})$ of size $(n-1)!$. Orbit-Stabiliser: $n!/(n-1)!=n$. ✓

Companion Theorem

The Orbit counting theorem complements Orbit-Stabiliser: while OS gives the size of an orbit, the orbit counting theorem gives the number of orbits.