Orbits and Stabilizers

Orbits, Stabilizers, and the Class Equation

Introduction

When a group acts on a set, it imposes a structure on that set, partitioning it into disjoint pieces. The study of these pieces provides a powerful counting tool that reveals deep information about the group itself.

Orbits and Stabilizers

Definitions

Definition 6.1: Let a group $G$ act on a set $X$ .

The orbit of an element $x \in X$ is the set of all elements in $X$ that $x$ can be moved to by the action of $G$ . It is denoted ${Orb}_{G} (x)$ or $G \cdot x$ :
${Orb}_{G} (x) = {g \cdot x ∣ g \in G}$
The stabilizer of an element $x \in X$ is the set of all elements in $G$ that fix $x$ . It is a subgroup of $G$ , denoted ${Stab}_{G} (x)$ or $G_{x}$ :
${Stab}_{G} (x) = {g \in G ∣ g \cdot x = x}$

Key Properties

Partition Property: The orbits form a partition of the set $X$ . This means that every element of $X$ belongs to exactly one orbit, and any two orbits are either identical or disjoint.
Subgroup Property: The stabilizer of any element is a subgroup of $G$ .
Relationship: These two concepts are fundamentally linked by one of the most useful theorems in group theory.

The Orbit-Stabilizer Theorem

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.

The Class Equation

A particularly insightful application of the Orbit-Stabilizer Theorem arises when a group $G$ acts on itself by conjugation. The action is defined by $g \cdot x = g x g^{- 1}$ for $g, x \in G$ .

Conjugacy Classes

The orbit of an element $x$ under this action is its conjugacy class, denoted $cl (x) = {g x g^{- 1} ∣ g \in G}$ .
The stabilizer of $x$ is the set of elements that commute with $x$ , known as the centralizer of $x$ , denoted $C_{G} (x) = {g \in G ∣ g x = x g}$ .

Applying the Orbit-Stabilizer Theorem

Applying the Orbit-Stabilizer Theorem to this action yields $| cl (x) | = [G : C_{G} (x)]$ .

Partitioning the Group

The set $G$ is partitioned by its conjugacy classes. We can separate the elements whose conjugacy class has size 1. These are the elements for which $g x g^{- 1} = x$ for all $g \in G$ , which is precisely the definition of the center of the group, $Z (G)$ .

The Class Equation

Summing the sizes of all the distinct conjugacy classes gives the order of the group:

| G | = \sum_{i = 1}^{k} | cl (x_{i}) |

where $x_{1}, \dots, x_{k}$ are representatives from each distinct conjugacy class. By separating the center, we arrive at the Class Equation:

| G | = | Z (G) | + \sum_{j = 1}^{m} [G : C_{G} (y_{j})]

where $y_{1}, \dots, y_{m}$ are representatives from the distinct conjugacy classes of size greater than 1.

Applications of the Class Equation

Application 1: p-Groups Have Non-trivial Centers

The Class Equation is a powerful numerical tool. Since each term $[G : C_{G} (y_{j})]$ must divide $| G |$ , it imposes strong constraints on the structure of a finite group.

A famous consequence is that any group whose order is a power of a prime $p$ (a p-group) must have a non-trivial center. This is because $| G |$ and each index $[G : C_{G} (y_{j})]$ are powers of $p$ , so for the equation to balance, $| Z (G) |$ must also be divisible by $p$ .

Application 2: Understanding Group Structure

The Class Equation helps us understand the internal structure of groups by revealing information about conjugacy classes and centralizers.

Application 3: Proving Simplicity

The Class Equation can be used to prove that certain groups are simple by showing that they cannot have non-trivial normal subgroups.

Examples

Example 1: Class Equation for $S_{3}$

Consider the symmetric group $S_{3}$ of order 6. The conjugacy classes are:

${e}$ (size 1)
${(1, 2), (1, 3), (2, 3)}$ (size 3)
${(1, 2, 3), (1, 3, 2)}$ (size 2)

The center $Z (S_{3}) = {e}$ , so the class equation is:

6 = 1 + 3 + 2

Example 2: Class Equation for $D_{4}$

Consider the dihedral group $D_{4}$ of order 8. The conjugacy classes are:

${e}$ (size 1)
${r^{2}}$ (size 1)
${r, r^{3}}$ (size 2)
${s, s r^{2}}$ (size 2)
${s r, s r^{3}}$ (size 2)

The center $Z (D_{4}) = {e, r^{2}}$ , so the class equation is:

8 = 2 + 2 + 2 + 2

Example 3: p-Group Example

Consider a group $G$ of order $p^{2}$ where $p$ is prime. By the class equation, $| Z (G) |$ must be divisible by $p$ . Since $Z (G) \leq G$ , we have $| Z (G) | = p$ or $| Z (G) | = p^{2}$ . If $| Z (G) | = p^{2}$ , then $G$ is abelian. If $| Z (G) | = p$ , then $G / Z (G)$ has order $p$ and is therefore cyclic, which implies that $G$ is abelian. Therefore, every group of order $p^{2}$ is abelian.

Burnside's Lemma

Another important application of group actions is Burnside's Lemma (also known as the Cauchy-Frobenius Lemma), which counts the number of orbits.

Statement

Theorem 6.3 (Burnside's Lemma): Let a finite group $G$ act on a finite set $X$ . The number of orbits is equal to the average number of fixed points:

Number of orbits = \frac{1}{| G |} \sum_{g \in G} | Fix (g) |

where $Fix (g) = {x \in X ∣ g \cdot x = x}$ is the set of elements fixed by $g$ .

Applications

Burnside's Lemma is particularly useful in combinatorics for counting objects up to symmetry. For example, it can be used to count:

The number of distinct colorings of a cube with $n$ colors
The number of distinct necklaces with $n$ beads of different colors
The number of distinct arrangements of objects under symmetry

Examples of Burnside's Lemma

Example 1: Counting Cube Colorings

How many distinct ways can we color the faces of a cube with 2 colors (red and blue)?

The group of symmetries of a cube has order 24. We need to count the fixed points of each symmetry:

Identity: fixes all $2^{6} = 64$ colorings
Rotations by $90 °$ about face axes: fix $2^{3} = 8$ colorings each
Rotations by $180 °$ about face axes: fix $2^{4} = 16$ colorings each
Rotations by $120 °$ about vertex axes: fix $2^{2} = 4$ colorings each
Rotations by $180 °$ about edge axes: fix $2^{3} = 8$ colorings each

By Burnside's Lemma:

Number of orbits = \frac{1}{24} (64 + 6 \cdot 8 + 3 \cdot 16 + 8 \cdot 4 + 6 \cdot 8) = \frac{1}{24} (64 + 48 + 48 + 32 + 48) = \frac{240}{24} = 10

So there are 10 distinct colorings.

Summary

The study of orbits and stabilizers provides powerful tools for understanding group actions and their applications. The Orbit-Stabilizer Theorem relates the size of orbits to the index of stabilizers, while the Class Equation reveals the internal structure of groups through their conjugacy classes. Burnside's Lemma provides a method for counting objects up to symmetry.

These concepts are fundamental to group theory and have applications throughout mathematics, from combinatorics to geometry to number theory.

Orbits, Stabilizers, and the Class Equation

Introduction