Commutator Subgroups

Introduction

The commutator subgroup measures how far a group is from being abelian. It is a fundamental concept that leads to the definitions of solvable and nilpotent groups—classes of groups that are "less complex" than simple groups and can be constructed from abelian groups.

Definition of Commutator

Definition 8.1: For two elements $g, h$ in a group $G$ , their commutator is the element $[g, h] = g^{- 1} h^{- 1} g h$ . The commutator is equal to the identity if and only if $g$ and $h$ commute.

Commutator Subgroup

Definition 8.2: The commutator subgroup (or derived subgroup) of $G$ , denoted $G^{'}$ or $[G, G]$ , is the subgroup generated by all commutators in $G$ :

G^{'} = ⟨ [g, h] ∣ g, h \in G ⟩

Properties

Normal Subgroup

The commutator subgroup $G^{'}$ is always a normal subgroup of $G$ .

Abelianization

The key property of $G^{'}$ is that the quotient group $G / G^{'}$ is the "largest" abelian quotient of $G$ . More precisely, a quotient group $G / N$ is abelian if and only if $N$ contains the commutator subgroup $G^{'}$ . The group $G / G^{'}$ is called the abelianization of $G$ .

Characterization

The commutator subgroup is the smallest normal subgroup $N$ of $G$ such that $G / N$ is abelian.

Derived Series

This process can be iterated to form the derived series of a group:

G^{(0)} = G, G^{(1)} = [G, G], G^{(2)} = [G^{(1)}, G^{(1)}], \dots, G^{(n + 1)} = [G^{(n)}, G^{(n)}]

This gives a descending chain of normal subgroups: $G ⊵ G^{(1)} ⊵ G^{(2)} ⊵ \dots$ .

Examples

Example 1: Abelian Groups

For an abelian group $G$ , we have $G^{'} = {e}$ , since all commutators are trivial.

Example 2: Dihedral Groups

In the dihedral group $D_{4}$ :

$D_{4}^{'} = ⟨ r^{2} ⟩$ (the subgroup generated by the rotation by 180°)
$D_{4}^{″} = {e}$

Example 3: Symmetric Groups

$S_{3}^{'} = A_{3}$
$S_{4}^{'} = A_{4}$
$S_{5}^{'} = A_{5}$

Example 4: Quaternion Group

In the quaternion group $Q_{8}$ :

$Q_{8}^{'} = ⟨ - 1 ⟩$

Applications

Application 1: Solvable Groups

The commutator subgroup is fundamental to the definition of solvable groups, which are groups whose derived series terminates in the trivial subgroup.

Application 2: Group Classification

The commutator subgroup helps classify groups by measuring their "non-abelianness."

Application 3: Galois Theory

The commutator subgroup is important in Galois theory, particularly in understanding solvable groups and their connection to polynomial equations.

Perfect Groups

A group $G$ is perfect if $G = G^{'}$ (i.e., $G$ equals its own commutator subgroup). Perfect groups are important because they cannot be "abelianized" further. Examples include:

The alternating groups $A_{n}$ for $n \geq 5$
The special linear groups $SL (n, F)$ for $n \geq 2$ and most fields $F$