Group Homomorphisms

Introduction

A homomorphism is a map between groups that preserves the group structure. Homomorphisms are fundamental to understanding the relationships between groups and are essential for the isomorphism theorems.

Definition

A homomorphism is a map $\varphi :G\to H$ between two groups that preserves the group structure, meaning $\varphi (g_1{g}_2)=\varphi (g_1)\varphi (g_2)$ for all $g_1,g_2\in G$.

Important Sets Associated with Homomorphisms

Two important sets associated with a homomorphism are:

1. Kernel

The kernel of $\varphi$ is the set of elements in $G$ that map to the identity in $H$:

Properties:

• The kernel is always a normal subgroup of $G$
• $\varphi$ is injective if and only if $\ker (\varphi )=\{e_G\}$

2. Image

The image of $\varphi$ is the set of elements in $H$ that are mapped to by some element in $G$:

Properties:

• The image is always a subgroup of $H$
• $\varphi$ is surjective if and only if $\text{Im}(\varphi )=H$

Examples

Example 1: Homomorphism from $\mathbb{Z}$ to ${\mathbb{Z}}_6$

Consider the homomorphism $\varphi :\mathbb{Z}\to {\mathbb{Z}}_6$ defined by $\varphi (n)=nmod6$.

• Kernel: $\ker (\varphi )=6\mathbb{Z}=\{0,\pm 6,\pm 12,\dots \}$
• Image: $\text{Im}(\varphi )={\mathbb{Z}}_6$
• Properties: Surjective but not injective

Example 2: Homomorphism from ${\mathbb{R}}^{\ast }$ to ${\mathbb{R}}^{\ast }$

Consider the homomorphism $\varphi :{\mathbb{R}}^{\ast }\to {\mathbb{R}}^{\ast }$ defined by $\varphi (x)=x^2$.

• Kernel: $\ker (\varphi )=\{\pm 1\}$
• Image: $\text{Im}(\varphi )={\mathbb{R}}^{+}$ (positive real numbers)
• Properties: Neither injective nor surjective

Example 3: Homomorphism from $S_n$ to $\{\pm 1\}$

Consider the sign homomorphism $\text{sgn}:S_n\to \{\pm 1\}$.

• Kernel: $\ker (\text{sgn})=A_n$ (alternating group)
• Image: $\text{Im}(\text{sgn})=\{\pm 1\}$
• Properties: Surjective but not injective for $n\ge 3$

Example 4: Homomorphism from $G{L}_n(\mathbb{R})$ to ${\mathbb{R}}^{\ast }$

Consider the determinant homomorphism $det:G{L}_n(\mathbb{R})\to {\mathbb{R}}^{\ast }$.

• Kernel: $\ker (det)=S{L}_n(\mathbb{R})$ (special linear group)
• Image: $\text{Im}(det)={\mathbb{R}}^{\ast }$
• Properties: Surjective but not injective

Properties

Basic Properties

1. Identity preservation: $\varphi (e_G)=e_H$
2. Inverse preservation: $\varphi (g^{-1})=\varphi (g{)}^{-1}$
3. Power preservation: $\varphi (g^n)=\varphi (g{)}^n$ for all $n\in \mathbb{Z}$

Composition

If $\varphi :G\to H$ and $\psi :H\to K$ are homomorphisms, then $\psi \circ \varphi :G\to K$ is also a homomorphism.

Restriction

If $\varphi :G\to H$ is a homomorphism and $S$ is a subgroup of $G$, then the restriction $\varphi {|}_S:S\to H$ is a homomorphism.

Types of Homomorphisms

Monomorphism

A homomorphism that is injective (one-to-one).

Epimorphism

A homomorphism that is surjective (onto).

Isomorphism

A homomorphism that is bijective (both injective and surjective).

Endomorphism

A homomorphism from a group to itself.

Automorphism

An isomorphism from a group to itself.

Applications

Application 1: Understanding Group Structure

Homomorphisms help us understand the structure of groups by relating them to other groups.

Application 2: First Isomorphism Theorem

Homomorphisms are essential for the first isomorphism theorem, which states that $G/\ker (\varphi )\cong \text{Im}(\varphi )$.

Application 3: Group Actions

Homomorphisms are fundamental to the study of group actions, where a group acts on a set.

Application 4: Representation Theory

Homomorphisms into matrix groups are the basis of representation theory.

Related Concepts

• Isomorphism Theorems
• Normal Subgroups
• Quotient Groups
• Group Actions
• Automorphisms