Stabilizers

Introduction

When a group acts on a set, the stabilizer of an element consists of all group elements that fix that element. Stabilizers are fundamental to understanding group actions and provide crucial information about the group's structure.

Definition

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

Subgroup Property

The stabilizer of any element is a subgroup of $G$ . This follows from:

Closure: If $g, h \in {Stab}_{G} (x)$ , then $(g h) \cdot x = g \cdot (h \cdot x) = g \cdot x = x$
Identity: $e \cdot x = x$ , so $e \in {Stab}_{G} (x)$
Inverses: If $g \in {Stab}_{G} (x)$ , then $g^{- 1} \cdot x = g^{- 1} \cdot (g \cdot x) = (g^{- 1} g) \cdot x = e \cdot x = x$

Relationship to Orbits

Stabilizers are fundamentally linked to orbits through the Orbit-Stabilizer Theorem.

Examples

Example 1: Natural Action of $S_{3}$

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

The stabilizer of 1 is ${Stab}_{S_{3}} (1) = {e, (2, 3)}$
The stabilizer of 2 is ${Stab}_{S_{3}} (2) = {e, (1, 3)}$
The stabilizer of 3 is ${Stab}_{S_{3}} (3) = {e, (1, 2)}$

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

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

The stabilizer of a vertex consists of the identity and the reflection across the diagonal through that vertex
${Stab}_{D_{4}} (v) = {e, reflection through v}$

Example 3: Conjugation Action

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

The stabilizer of an element $x$ is its centralizer: $C_{G} (x) = {g \in G ∣ g x = x g}$
Elements in the centralizer commute with $x$

Example 4: Left Multiplication Action

When a group $G$ acts on itself by left multiplication:

The stabilizer of any element is trivial: ${Stab}_{G} (x) = {e}$
This is why the action is free

Applications

Application 1: Understanding Group Structure

Stabilizers help us understand the internal structure of groups by revealing which elements fix particular points.

Application 2: Orbit-Stabilizer Theorem

Stabilizers are essential for the Orbit-Stabilizer Theorem, which relates orbit size to stabilizer index.

Application 3: Class Equation

When studying conjugation actions, stabilizers (centralizers) lead to the Class Equation.

Application 4: Counting Problems

Stabilizers are crucial in counting problems involving symmetry, particularly in combinatorics.

Special Cases

Trivial Stabilizer

If ${Stab}_{G} (x) = {e}$ for all $x \in X$ , the action is called free.

Full Stabilizer

If ${Stab}_{G} (x) = G$ for some $x \in X$ , then $x$ is a fixed point of the action.

Normal Subgroup

If $N$ is a normal subgroup of $G$ , then $N$ is the stabilizer of the identity coset in the action of $G$ on $G / N$ .