First Isomorphism Theorem

Introduction

The First Isomorphism Theorem is arguably the most important theorem in group theory. It provides a direct link between homomorphisms and quotient groups, showing that factoring a group by the kernel of a map precisely captures the structure of the map's image.

Statement

Theorem 3.1 (First Isomorphism Theorem): Let $ϕ : G \to H$ be a group homomorphism. Then the quotient group $G / \ker (ϕ)$ is isomorphic to the image of $ϕ$ .

G / \ker (ϕ) ≅ Im (ϕ)

Proof Sketch

The proof involves constructing an explicit isomorphism $ψ : G / \ker (ϕ) \to Im (ϕ)$ defined by $ψ (g \ker (ϕ)) = ϕ (g)$ .

Steps:

Well-defined: Show that if $g \ker (ϕ) = g^{'} \ker (ϕ)$ , then $ϕ (g) = ϕ (g^{'})$
Homomorphism: Show that $ψ$ preserves the group operation
Injective: Show that $\ker (ψ) = {\ker (ϕ)}$
Surjective: Show that every element in $Im (ϕ)$ is in the image of $ψ$

Examples

Example 1: Integers Modulo n

Consider the homomorphism $ϕ : Z \to Z_{6}$ defined by $ϕ (n) = n mod 6$ .

$\ker (ϕ) = 6 Z$
$Im (ϕ) = Z_{6}$
By the First Isomorphism Theorem: $Z / 6 Z ≅ Z_{6}$

Example 2: Real Numbers

Consider the homomorphism $ϕ : R^{*} \to R^{*}$ defined by $ϕ (x) = x^{2}$ .

$\ker (ϕ) = {\pm 1}$
$Im (ϕ) = R^{+}$ (positive real numbers)
By the First Isomorphism Theorem: $R^{*} / {\pm 1} ≅ R^{+}$

Example 3: Symmetric Groups

Consider the sign homomorphism $sgn : S_{n} \to {\pm 1}$ .

$\ker (sgn) = A_{n}$ (alternating group)
$Im (sgn) = {\pm 1}$
By the First Isomorphism Theorem: $S_{n} / A_{n} ≅ {\pm 1}$

Example 4: Matrix Groups

Consider the determinant homomorphism $det : G L_{n} (R) \to R^{*}$ .

$\ker (det) = S L_{n} (R)$ (special linear group)
$Im (det) = R^{*}$
By the First Isomorphism Theorem: $G L_{n} (R) / S L_{n} (R) ≅ R^{*}$

Applications

Application 1: Understanding Group Structure

The First Isomorphism Theorem helps us understand the structure of groups by relating them to simpler groups through homomorphisms.

Example: If we have a homomorphism $ϕ : G \to H$ with kernel $K$ , then $G$ is "built" from $K$ and $H$ in a precise way.

Application 2: Proving Isomorphisms

The theorem provides a powerful tool for proving that two groups are isomorphic without having to construct an explicit isomorphism.

Example: To show that $Z / n Z ≅ Z_{n}$ , we can use the homomorphism $ϕ : Z \to Z_{n}$ defined by $ϕ (k) = k mod n$ .

Application 3: Classification of Groups

The theorem is essential for understanding the structure of groups and their classification.

Example: Understanding the structure of finite abelian groups through their decomposition into cyclic groups.

Application 4: Galois Theory

The First Isomorphism Theorem is fundamental in Galois theory, where it helps understand the relationship between field extensions and their Galois groups.

Special Cases

Case 1: Injective Homomorphism

If $ϕ : G \to H$ is injective, then $\ker (ϕ) = {e}$ , so $G / {e} ≅ G ≅ Im (ϕ)$ . This shows that $G$ is isomorphic to a subgroup of $H$ .

Case 2: Surjective Homomorphism

If $ϕ : G \to H$ is surjective, then $Im (ϕ) = H$ , so $G / \ker (ϕ) ≅ H$ . This shows that $H$ is isomorphic to a quotient of $G$ .

Case 3: Isomorphism

If $ϕ : G \to H$ is an isomorphism, then $\ker (ϕ) = {e}$ and $Im (ϕ) = H$ , so $G / {e} ≅ G ≅ H$ .

Second Isomorphism Theorem

Relates the quotient $(S N) / N$ to $S / (S \cap N)$ for subgroups $S$ and normal subgroups $N$ .

Third Isomorphism Theorem

Provides a "cancellation" rule for quotients: $(G / N) / (K / N) ≅ G / K$ .

Summary

The First Isomorphism Theorem is a fundamental result that connects homomorphisms, normal subgroups, and quotient groups. It provides a powerful tool for understanding group structure and proving isomorphisms. The theorem is essential for the deeper study of group theory and its applications throughout mathematics.