Orbits

Introduction

When a group acts on a set, it partitions the set into disjoint pieces called orbits. Understanding orbits is fundamental to understanding group actions and their applications.

Definition

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 ${\text{Orb}}_G(x)$ or $G\cdot x$:

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
• Any two orbits are either identical or disjoint

Equivalence Relation

The relation $x\sim y$ if and only if $y\in {\text{Orb}}_G(x)$ is an equivalence relation on $X$, and the equivalence classes are precisely the orbits.

Transitive Actions

If the action is transitive, there is only one orbit: the entire set $X$.

Examples

Example 1: Natural Action of $S_3$

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)
• ${\text{Orb}}_{S_3}(1)={\text{Orb}}_{S_3}(2)={\text{Orb}}_{S_3}(3)=\{1,2,3\}$

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)
• ${\text{Orb}}_{D_4}(v)=\{\text{all vertices}\}$ for any vertex $v$

Example 3: Conjugation Action

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

• The orbit of an element $x$ is its conjugacy class: $\text{cl}(x)=\{gx{g}^{-1}\mid g\in G\}$
• Different elements may have different sized orbits

Example 4: Action on Cosets

Let $H$ be a subgroup of $G$. When $G$ acts on the set of left cosets $G/H$ by left multiplication:

• The orbit of any coset is the entire set $G/H$ (transitive action)

Applications

Application 1: Understanding Group Structure

Orbits help us understand how a group acts on a set and reveal information about the group's structure.

Application 2: Counting Problems

Orbits are fundamental in counting problems involving symmetry, particularly in combinatorics.

Application 3: Geometry

Orbits represent the "paths" that elements can follow under the group action, which is important in geometry and symmetry.

Application 4: Representation Theory

Orbits are important in representation theory, where they correspond to irreducible representations.

Related Concepts

• Stabilizers
• Orbit-Stabilizer Theorem
• Group Actions
• Conjugacy Classes
• Transitive Actions