Second Isomorphism Theorem

Introduction

The Second Isomorphism Theorem, also known as the "Diamond Isomorphism Theorem" due to the shape of the corresponding subgroup lattice diagram, relates the quotient of a product of subgroups to the quotient of their intersection.

Statement

Theorem 3.2 (Second Isomorphism Theorem): Let $G$ be a group, $S$ a subgroup of $G$ , and $N$ a normal subgroup of $G$ . Then the product $S N = {s n ∣ s \in S, n \in N}$ is a subgroup of $G$ , the intersection $S \cap N$ is a normal subgroup of $S$ , and the following isomorphism holds:

(S N) / N ≅ S / (S \cap N)

Proof Sketch

The proof involves several steps:

Show $S N$ is a subgroup: Use the subgroup test to verify that $S N$ is closed under the group operation and inverses.
Show $S \cap N$ is normal in $S$ : For any $s \in S$ and $x \in S \cap N$ , show that $s x s^{- 1} \in S \cap N$ .
Construct the isomorphism: Define a homomorphism $ϕ : S \to (S N) / N$ by $ϕ (s) = s N$ and show that its kernel is $S \cap N$ .
Apply the First Isomorphism Theorem: Since $\ker (ϕ) = S \cap N$ and $Im (ϕ) = (S N) / N$ , we get $S / (S \cap N) ≅ (S N) / N$ .

Examples

Example 1: Symmetric Groups

Let $G = S_{4}$ , $S = ⟨ (1, 2, 3) ⟩$ (cyclic subgroup of order 3), and $N = A_{4}$ (alternating group).

$S N = S_{4}$ (since $S_{4} = A_{4} \cup A_{4} (1, 2, 3)$ )
$S \cap N = {e}$
By the Second Isomorphism Theorem: $S_{4} / A_{4} ≅ ⟨ (1, 2, 3) ⟩ / {e} ≅ Z_{3}$

Example 2: Dihedral Groups

Let $G = D_{6}$ , $S = ⟨ r^{2} ⟩$ (subgroup of rotations by multiples of $120 °$ ), and $N = ⟨ r^{3} ⟩$ (subgroup of rotations by multiples of $180 °$ ).

$S N = ⟨ r ⟩$ (all rotations)
$S \cap N = {e}$
By the Second Isomorphism Theorem: $⟨ r ⟩ / ⟨ r^{3} ⟩ ≅ ⟨ r^{2} ⟩ / {e} ≅ Z_{3}$

Example 3: Abelian Groups

Let $G = Z_{12}$ , $S = ⟨ 2 ⟩ = {0, 2, 4, 6, 8, 10}$ , and $N = ⟨ 3 ⟩ = {0, 3, 6, 9}$ .

$S N = Z_{12}$ (since $gcd (2, 3) = 1$ )
$S \cap N = ⟨ 6 ⟩ = {0, 6}$
By the Second Isomorphism Theorem: $Z_{12} / ⟨ 3 ⟩ ≅ ⟨ 2 ⟩ / ⟨ 6 ⟩ ≅ Z_{3}$

Applications

Application 1: Understanding Subgroup Structure

The Second Isomorphism Theorem helps us understand how subgroups interact with normal subgroups and their quotients.

Application 2: Index Calculations

The theorem can be used to calculate indices: $[S N : N] = [S : S \cap N]$ .

Application 3: Proving Isomorphisms

The theorem provides a powerful tool for proving that two quotient groups are isomorphic.

Special Cases

Case 1: $S \cap N = {e}$

If $S \cap N = {e}$ , then $(S N) / N ≅ S$ . This means that $S$ is isomorphic to a subgroup of $G / N$ .

Case 2: $S$ is Normal

If $S$ is also normal, then $S N$ is normal and the theorem becomes a special case of the Third Isomorphism Theorem.

Case 3: $S \subseteq N$

If $S \subseteq N$ , then $S N = N$ and $S \cap N = S$ , so the theorem becomes the trivial statement $N / N ≅ S / S$ .

First Isomorphism Theorem

Relates $G / \ker (ϕ)$ to $Im (ϕ)$ for homomorphisms $ϕ : G \to H$ .

Third Isomorphism Theorem

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

Summary

The Second Isomorphism Theorem is a powerful tool for understanding the relationships between subgroups, normal subgroups, and their quotients. It is particularly useful when working with products of subgroups and provides a bridge between different quotient constructions.