Quotient Groups

Introduction

When a subgroup $N$ is normal, the set of its cosets, $G/N$, forms a group under a natural operation. This construction is fundamental to group theory and provides a powerful method for understanding group structure.

Definition

When a subgroup $N$ is normal, the set of its cosets, $G/N$, forms a group under the operation $(aN)(bN)=abN$. This group is called the quotient group or factor group of $G$ by $N$.

Group Structure

• Identity element: The coset $eN=N$
• Inverse: The inverse of $aN$ is $a^{-1}N$
• Operation: $(aN)(bN)=abN$

Construction and Interpretation

The construction of a quotient group is a fundamental method for simplifying a group. It effectively "collapses" or "factors out" the structure of the normal subgroup $N$, treating all elements of $N$ as equivalent to the identity.

Well-Defined Operation

The operation $(aN)(bN)=abN$ is well-defined because $N$ is normal. This means that if $aN=a^{\prime }N$ and $bN=b^{\prime }N$, then $abN=a^{\prime }{b}^{\prime }N$.

Key Example: Integers Modulo n

A key example is the group of integers modulo $n$, $(\mathbb{Z}/n\mathbb{Z},+)$, which is the quotient group of the integers $(\mathbb{Z},+)$ by the normal subgroup $n\mathbb{Z}$.

Structure

• Elements: $\{0+n\mathbb{Z},1+n\mathbb{Z},\dots ,(n-1)+n\mathbb{Z}\}$
• Operation: $(a+n\mathbb{Z})+(b+n\mathbb{Z})=(a+b)+n\mathbb{Z}$
• Order: $n$
• Isomorphism: $\mathbb{Z}/n\mathbb{Z}\cong {\mathbb{Z}}_n$

This construction is the foundation of modular arithmetic.

Examples

Example 1: Quotient Groups in Abelian Groups

In an abelian group, every subgroup is normal, so we can always form quotient groups.

Example: Consider $(\mathbb{Z},+)$ and the subgroup $6\mathbb{Z}$:

• $\mathbb{Z}/6\mathbb{Z}=\{0+6\mathbb{Z},1+6\mathbb{Z},2+6\mathbb{Z},3+6\mathbb{Z},4+6\mathbb{Z},5+6\mathbb{Z}\}$
• This is isomorphic to ${\mathbb{Z}}_6$

Example 2: Quotient Groups in $D_4$

Consider the normal subgroup $N=\{e,r^2\}$ in $D_4$:

• $D_4/N=\{N,rN,sN,srN\}$
• The operation is well-defined because $N$ is normal
• $D_4/N$ is a group of order 4

Example 3: Quotient Groups in $S_n$

The alternating group $A_n$ is a normal subgroup of $S_n$:

• $S_n/A_n=\{A_n,(1,2)A_n\}$
• This is isomorphic to $\{\pm 1\}$ under multiplication

Example 4: Quotient Groups in Matrix Groups

Consider the group $G{L}_n(\mathbb{R})$ and the normal subgroup $S{L}_n(\mathbb{R})$ (matrices with determinant 1):

• $G{L}_n(\mathbb{R})/S{L}_n(\mathbb{R})\cong {\mathbb{R}}^{\ast }$ (the multiplicative group of non-zero real numbers)
• The isomorphism is given by $A\cdot S{L}_n(\mathbb{R})\mapsto det(A)$

Properties

Order

If $G$ is finite, then $|G/N|=|G|/|N|=[G:N]$.

Abelian Quotients

If $G$ is abelian, then $G/N$ is abelian for any normal subgroup $N$.

Cyclic Quotients

If $G$ is cyclic, then $G/N$ is cyclic for any normal subgroup $N$.

Simple Groups

A group $G$ is simple if and only if its only normal subgroups are $\{e\}$ and $G$ itself.

Applications

Application 1: Group Structure

Quotient groups help us understand the structure of groups by "factoring out" normal subgroups.

Application 2: Homomorphisms

Quotient groups are essential for the first isomorphism theorem, which states that $G/\ker (\varphi )\cong \text{im}(\varphi )$ for any homomorphism $\varphi :G\to H$.

Application 3: Galois Theory

Quotient groups are fundamental in Galois theory, where they correspond to intermediate fields.

Application 4: Representation Theory

Quotient groups are important in representation theory and the study of group actions.

The Natural Homomorphism

For any normal subgroup $N$ of $G$, there is a natural homomorphism $\pi :G\to G/N$ defined by $\pi (g)=gN$. This homomorphism is surjective and has kernel $N$.

Properties

• $\pi$ is surjective
• $\ker (\pi )=N$
• $\text{im}(\pi )=G/N$

Related Concepts

• Normal Subgroups
• Cosets
• Group Homomorphisms
• Isomorphism Theorems
• Simple Groups