Orbit-Stabilizer Theorem

Introduction

The Orbit-Stabilizer Theorem is one of the most useful theorems in group theory. It provides a fundamental relationship between the size of orbits and the index of stabilizers, offering a powerful tool for counting and understanding group actions.

Statement

Theorem 6.2 (Orbit-Stabilizer Theorem): Let a finite group $G$ act on a set $X$ . For any $x \in X$ , the size of the orbit of $x$ is equal to the index of its stabilizer subgroup:

| {Orb}_{G} (x) | = \frac{| G |}{| {Stab}_{G} (x) |}

This theorem provides a powerful formula: $| G | = | {Orb}_{G} (x) | \cdot | {Stab}_{G} (x) |$ .

Proof Sketch

The proof uses the fact that there is a bijection between the orbit of $x$ and the set of left cosets of the stabilizer of $x$ . Specifically, the map $g \cdot x \mapsto g {Stab}_{G} (x)$ is a well-defined bijection.

Key Steps in the Proof

Define the map: $ϕ : {Orb}_{G} (x) \to G / {Stab}_{G} (x)$ by $ϕ (g \cdot x) = g {Stab}_{G} (x)$
Show it's well-defined: If $g_{1} \cdot x = g_{2} \cdot x$ , then $g_{1}^{- 1} g_{2} \in {Stab}_{G} (x)$ , so $g_{1} {Stab}_{G} (x) = g_{2} {Stab}_{G} (x)$
Show it's injective: If $ϕ (g_{1} \cdot x) = ϕ (g_{2} \cdot x)$ , then $g_{1} \cdot x = g_{2} \cdot x$
Show it's surjective: Every coset $g {Stab}_{G} (x)$ is the image of $g \cdot x$
Conclude: Since there's a bijection, $| {Orb}_{G} (x) | = [G : {Stab}_{G} (x)] = | G | / | {Stab}_{G} (x) |$

Examples

Example 1: Action of $S_{3}$ on ${1, 2, 3}$

Consider the natural action of $S_{3}$ on ${1, 2, 3}$ :

The orbit of any element is the entire set ${1, 2, 3}$ (the action is transitive)
The stabilizer of 1 is ${e, (2, 3)}$
By the orbit-stabilizer theorem: $| S_{3} | = 6 = 3 \cdot 2 = | Orb (1) | \cdot | Stab (1) |$

Example 2: Action of $D_{4}$ on Vertices

Consider the action of $D_{4}$ on the vertices of a square:

The orbit of any vertex is the entire set of vertices (transitive action)
The stabilizer of a vertex consists of the identity and the reflection across the diagonal through that vertex
By the orbit-stabilizer theorem: $| D_{4} | = 8 = 4 \cdot 2 = | Orb (v) | \cdot | Stab (v) |$

Example 3: Conjugation Action

When a group $G$ acts on itself by conjugation:

The orbit of an element $x$ is its conjugacy class: $cl (x) = {g x g^{- 1} ∣ g \in G}$
The stabilizer of $x$ is its centralizer: $C_{G} (x) = {g \in G ∣ g x = x g}$
The orbit-stabilizer theorem gives: $| cl (x) | = [G : C_{G} (x)]$

Applications

Application 1: Counting Orbits

The orbit-stabilizer theorem is often used to count the number of elements in orbits, which is useful in combinatorics and group theory.

Application 2: Understanding Group Structure

The theorem helps us understand the structure of groups by relating orbit sizes to subgroup indices.

Application 3: Class Equation

When a group acts on itself by conjugation, the orbit-stabilizer theorem leads to the class equation, which is fundamental for understanding the structure of finite groups.

Application 4: Burnside's Lemma

The orbit-stabilizer theorem is used in the proof of Burnside's Lemma, which counts the number of orbits.

Special Cases

Transitive Actions

If the action is transitive, then $| {Orb}_{G} (x) | = | X |$ for all $x \in X$ , so $| G | = | X | \cdot | {Stab}_{G} (x) |$ .

Free Actions

If the action is free, then $| {Stab}_{G} (x) | = 1$ for all $x \in X$ , so $| {Orb}_{G} (x) | = | G |$ .

Regular Actions

If the action is regular (transitive and free), then $| G | = | X |$ .