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

{Orb}_{G} (x) = {g \cdot x ∣ g \in G}

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 $y \in {Orb}_{G} (x)$ 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)
${Orb}_{S_{3}} (1) = {Orb}_{S_{3}} (2) = {Orb}_{S_{3}} (3) = {1, 2, 3}$

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)
${Orb}_{D_{4}} (v) = {all vertices}$ 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: $cl (x) = {g x g^{- 1} ∣ g \in G}$
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.