The Isomorphism Theorems — Mathematics Knowledge System

Overview

The isomorphism theorems are a set of foundational results, largely due to Emmy Noether, that describe precisely how quotient groups, subgroups, and homomorphisms relate to each other. The First Isomorphism Theorem says every homomorphism factors as an embedding of G/ker(φ) into the codomain, identifying the quotient with the image. The Second and Third build on this to relate subgroups of a quotient back to subgroups of the original group. Together they are the fundamental toolkit for understanding group structure by comparing a group to its quotients and images.

Intuition

The First Isomorphism Theorem says a homomorphism can't lose information beyond collapsing its kernel: once you quotient out exactly the elements that map to the identity, what's left is a perfect (isomorphic) copy of the image. The Third Isomorphism Theorem is the group-theoretic analogue of canceling fractions — 'dividing by N, then by K/N' is the same as 'dividing by K directly,' just like (a/n)/(k/n) = a/k. The Correspondence Theorem says quotienting by N doesn't create or destroy subgroup structure above N — it just relabels it.

Formal Definition

Definition

Let φ: G → H be a group homomorphism, N ⊴ G a normal subgroup, and A, B ≤ G.

G/\ker(\varphi) \cong \operatorname{im}(\varphi)

First Isomorphism Theorem

AB/B \cong A/(A \cap B) \quad \text{if } A \leq G,\ B \trianglelefteq G

Second (Diamond) Isomorphism Theorem

(G/N)/(K/N) \cong G/K \quad \text{for } N \trianglelefteq K \trianglelefteq G

Third Isomorphism Theorem

\{\text{subgroups of } G/N\} \longleftrightarrow \{\text{subgroups of } G \text{ containing } N\}

Fourth (Correspondence) Isomorphism Theorem

Worked Examples

The kernel is the set of even permutations, which is A₃ = {e, (123), (132)}.
$\ker(\varphi) = A_3,\ |A_3| = 3$
The image is all of {±1}, since both even and odd permutations exist in S₃.
$\operatorname{im}(\varphi) = \{+1,-1\} \cong \mathbb{Z}_2$
The First Isomorphism Theorem gives S₃/A₃ ≅ {±1}, and indeed |S₃|/|A₃| = 6/3 = 2 = |{±1}|.
$S_3/A_3 \cong \mathbb{Z}_2$

Answer: S₃/A₃ ≅ ℤ₂, matching the order count 6/3 = 2.

Practice Problems

Difficulty 6/10

Let φ: G → H be a homomorphism with G finite. Using the First Isomorphism Theorem, explain why |im(φ)| must divide |G|.

Difficulty 7/10

For the determinant homomorphism det: GL₂(ℝ) → ℝ*, identify ker(det) and describe what the First Isomorphism Theorem tells you.

Difficulty 6/10

State the Correspondence (Fourth Isomorphism) Theorem in your own words and give one consequence for finding all subgroups of ℤ/12ℤ.

Quiz

Summary

First Isomorphism Theorem: G/ker(φ) ≅ im(φ) for any group homomorphism φ.
Second (Diamond): AB/B ≅ A/(A∩B); Third: (G/N)/(K/N) ≅ G/K for N ⊴ K ⊴ G — quotients 'cancel' like fractions.
Correspondence Theorem: subgroups of G/N biject with subgroups of G containing N, preserving normality and index.

References

WebsiteWikipedia — Isomorphism theorems