Group Actions

Introduction

The concept of a group "acting" on a set is a powerful generalization of the idea that groups represent symmetries. It provides a framework for studying how group elements permute or transform other mathematical objects, leading to some of the most important theorems in the subject.

Definition

Formal Definition

A (left) group action of a group $G$ on a set $X$ is a map $G \times X \to X$ , denoted by $(g, x) \mapsto g \cdot x$ , that satisfies two axioms for all $g, h \in G$ and $x \in X$ :

Identity: $e \cdot x = x$
Compatibility: $g \cdot (h \cdot x) = (g h) \cdot x$

Alternative Definition

Equivalently, a group action is a homomorphism $ϕ : G \to Sym (X)$ , where $Sym (X)$ is the group of all permutations of the set $X$ . The kernel of this homomorphism, $\ker (ϕ) = {g \in G ∣ g \cdot x = x for all x \in X}$ , consists of the elements of $G$ that act trivially on $X$ .

Properties of Actions

Faithful Actions

The action is faithful if the kernel is trivial ( $\ker (ϕ) = {e}$ ). In this case, $G$ is isomorphic to a subgroup of $Sym (X)$ .

Transitive Actions

The action is transitive if for any two elements $x, y \in X$ , there exists a $g \in G$ such that $g \cdot x = y$ . This means the group can move any element to any other element.

Free Actions

The action is free if the stabilizer of every element is trivial, i.e., $g \cdot x = x$ implies $g = e$ .

Regular Actions

The action is regular if it is both transitive and free.

Examples

Example 1: Natural Action of $S_{n}$

The symmetric group $S_{n}$ acts on the set ${1, 2, \dots, n}$ . This is the natural action:

For $σ \in S_{n}$ and $i \in {1, 2, \dots, n}$ , we define $σ \cdot i = σ (i)$
This action is faithful and transitive

Example 2: Dihedral Group Action

The dihedral group $D_{n}$ acts on the set of vertices of a regular $n$ -gon:

Rotations and reflections permute the vertices
This action is faithful and transitive

Example 3: Left Multiplication Action

Any group $G$ acts on itself by left multiplication: $(g, x) \mapsto g x$ . This action is always faithful and regular (transitive and free).

Example 4: Conjugation Action

Any group $G$ acts on itself by conjugation: $(g, x) \mapsto g x g^{- 1}$ . This action is fundamental for studying the internal structure of the group.

Example 5: Action on Cosets

Let $H$ be a subgroup of $G$ . Then $G$ acts on the set of left cosets $G / H$ by left multiplication: $(g, a H) \mapsto g a H$ .

Applications

Application 1: Cayley's Theorem

The left multiplication action of a group on itself provides a proof of Cayley's theorem: every group is isomorphic to a subgroup of a symmetric group.

Application 2: Understanding Group Structure

Group actions help us understand the structure of groups by studying how they act on various sets.

Application 3: Combinatorics

Group actions are fundamental in combinatorics, particularly in counting problems involving symmetry.

Application 4: Geometry

Group actions are essential in geometry, where they represent symmetries of geometric objects.