Proof of the Orbit-Stabilizer Theorem

The orbit-stabilizer theorem is a fundamental result connecting the size of a group to the orbit and stabilizer of an element under a group action. It is the key ingredient in proving Burnside's lemma.

Statement

Theorem8.3Orbit-Stabilizer Theorem

Let $G$ be a finite group acting on a set $X$ . For any $x \in X$ : $|G| = |\operatorname{Orb}(x)| \cdot |\operatorname{Stab}(x)|.$

Proof

Define the map $\phi: G/\operatorname{Stab}(x) \to \operatorname{Orb}(x)$ by $\phi(g \cdot \operatorname{Stab}(x)) = g \cdot x$ .

Well-defined: If $g_1 \operatorname{Stab}(x) = g_2 \operatorname{Stab}(x)$ , then $g_2^{-1}g_1 \in \operatorname{Stab}(x)$ , so $g_2^{-1}g_1 \cdot x = x$ , giving $g_1 \cdot x = g_2 \cdot x$ .

Injective: If $g_1 \cdot x = g_2 \cdot x$ , then $g_2^{-1}g_1 \cdot x = x$ , so $g_2^{-1}g_1 \in \operatorname{Stab}(x)$ , and $g_1 \operatorname{Stab}(x) = g_2 \operatorname{Stab}(x)$ .

Surjective: For any $y \in \operatorname{Orb}(x)$ , there exists $g \in G$ with $g \cdot x = y$ , and $\phi(g \operatorname{Stab}(x)) = y$ .

Therefore $\phi$ is a bijection, so $|\operatorname{Orb}(x)| = |G/\operatorname{Stab}(x)| = |G|/|\operatorname{Stab}(x)|$ by Lagrange's theorem. $\square$

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Applications

ExampleRotations of the Tetrahedron

The rotation group of the regular tetrahedron has order 12. A face of the tetrahedron has stabilizer of order 3 (the three rotations fixing that face). By the orbit-stabilizer theorem, the face orbit has size $12/3 = 4$ , confirming that all 4 faces form a single orbit. Similarly, an edge has stabilizer of order 2 (identity and the rotation flipping the edge), so the edge orbit has size $12/2 = 6$ , matching the 6 edges.

RemarkGeneralizations

The orbit-stabilizer theorem generalizes in several directions:

Burnside's lemma follows by summing $1/|\operatorname{Orb}(x)|$ over all $x$ .
For continuous groups, the theorem becomes $\dim G = \dim(\text{orbit}) + \dim(\text{stabilizer})$ .
In the context of categories, it generalizes to the groupoid cardinality formula.

ExampleConjugacy Classes as Orbits

A group $G$ acts on itself by conjugation: $g \cdot x = gxg^{-1}$ . The orbits are the conjugacy classes, and $\operatorname{Stab}(x) = C_G(x)$ , the centralizer of $x$ . By the orbit-stabilizer theorem: $|\text{conjugacy class of } x| = \frac{|G|}{|C_G(x)|} = [G : C_G(x)].$ This is the basis of the class equation: $|G| = |Z(G)| + \sum_{i} [G : C_G(x_i)]$ , summing over representatives of non-central conjugacy classes.

RemarkCounting Orbits with Weights

The orbit-stabilizer theorem leads to the weighted Burnside lemma: if each element $x \in X$ has weight $w(x)$ constant on orbits, the total weight of orbits is: $\sum_{\text{orbits}} w = \frac{1}{|G|} \sum_{g \in G} \sum_{\substack{x \in X \\ g \cdot x = x}} w(x).$ This is the foundation for the Pólya Enumeration Theorem.