← Graph View All Domains

A topic in the Open Knowledge Graph — a free, open map of 15,290 topics and the order to learn them in.

Second Isomorphism Theorem for Groups

Graduate Depth 98 in the knowledge graph ☐ I know this ☆ Set as goal

31topics build on this

411prerequisites beneath it

See this on the map →

First Isomorphism Theorem for Groups Fundamental Theorem of Galois Theory +2 more→→Direct Products of Groups

Core Idea

Let H be a subgroup and N a normal subgroup of G. Then HN is a subgroup, N is normal in HN, and (HN)/N ≅ H/(H ∩ N).

Explainer

The Second Isomorphism Theorem describes what happens when a subgroup H and a normal subgroup N interact inside a larger group G. You already know from the First Isomorphism Theorem that quotients and homomorphisms are deeply linked — this theorem extends that insight to a finer structural picture involving how two subgroups overlap and combine.

The setup: H is any subgroup of G and N is a normal subgroup. The theorem says three things at once. First, HN = {hn : h ∈ H, n ∈ N} is itself a subgroup of G (this wouldn't hold in general without N being normal). Second, N is normal inside HN (it was normal in all of G, so it's certainly normal in the smaller group HN). Third, and most importantly, there's an isomorphism (HN)/N ≅ H/(H ∩ N). The key to proving this is the map φ: H → (HN)/N defined by φ(h) = hN. This map is a surjective homomorphism, and its kernel is {h ∈ H : hN = N} = H ∩ N. The First Isomorphism Theorem then delivers the result.

A concrete example: take G = Z₁₂, H = ⟨4⟩ = {0, 4, 8}, N = ⟨3⟩ = {0, 3, 6, 9}. Then HN contains both 4 and 3, and since gcd(3, 4) = 1 in Z₁₂ these generate all of Z₁₂, so HN = Z₁₂. Meanwhile H ∩ N = {0} (the two cyclic subgroups share only the identity). The theorem says Z₁₂/{0, 3, 6, 9} ≅ {0, 4, 8}/{0}, i.e., Z₃ ≅ Z₃ — the sizes match (4 elements on the left, 3 elements on the right... wait, Z₁₂/N has 3 cosets). The theorem keeps the "sizes" consistent: |HN|/|N| = |H|/|H ∩ N|.

The theorem is sometimes called the diamond isomorphism theorem because the four groups N, H, HN, and H ∩ N form a diamond shape in the subgroup lattice. The isomorphism (HN)/N ≅ H/(H ∩ N) says the "ratio" between HN and N equals the "ratio" between H and H ∩ N — a beautiful symmetry in the lattice structure of the group. This perspective becomes especially powerful when studying the correspondence theorem and the structure of quotient groups in more advanced algebra.

Practice Questions 5 questions

Prerequisite Chain

Understanding Zero → The Number Zero → Counting to Five → Counting to 10 → Counting to 20 → Counting a Set of Objects Up to 20 → Cardinality: The Last Number Counted → Matching Numerals to Quantities → Subitizing Small Quantities → Addition Within 10 → Number Bonds to 10 → Addition Within 20 → Doubles and Near Doubles → Doubles Facts Within 10 → Near Doubles Facts Within 20 → Mental Math Strategies for Addition → Mental Math: Adding and Subtracting Tens → Addition Within 100 → Repeated Addition as Multiplication → Multiplication as Equal Groups → Multiplication: Arrays → Basic Multiplication Facts (0s, 1s, 2s, 5s, 10s) → Multiplication Facts Within 100 → Division as Equal Sharing → Division as Grouping (Measurement Division) → Division: Grouping (Repeated Subtraction) Model → Division: Fair Sharing Model → Division as Equal Sharing → Division as Grouping → Basic Division Facts → Division Facts Within 100 → Multiplication and Division Fact Families → Relationship Between Multiplication and Division → Division Facts as Inverse of Multiplication → Remainders and Quotients in Division → Division Word Problems → Multi-Step Word Problems → Solving Multi-Step Word Problems → Multiplication Word Problems → Division Word Problems → Introduction to Long Division → Factors and Multiples → Prime and Composite Numbers → Equivalent Fractions → Relating Fractions and Decimals → Decimal Place Value → Integers and the Number Line → Comparing and Ordering Integers → Absolute Value → Adding Integers → Subtracting Integers → Multiplying Integers → Introduction to Exponents → Order of Operations → Integer Order of Operations → Variable Expressions → The Distributive Property → Variables and Expressions Review → Introduction to Polynomials → Adding and Subtracting Polynomials → Multiplying Polynomials → Factorial → Permutations → Combinations → Counting Principles: Addition and Multiplication Rules → Introduction to Graph Theory → Propositional Logic Foundations → Logical Equivalences → Set Operations: Union, Intersection, and Complement → Proof by Cases → Proving by Cases and Exhaustion → Vacuous Truth and Trivial Cases → Proof by Cases (Proof by Exhaustion) → Mathematical Induction → Binary Operations and Algebraic Structures → Group Definition and Examples → Basic Properties of Groups → Group Homomorphisms → Group Isomorphisms → Cayley's Theorem → Cosets and Lagrange's Theorem → Normal Subgroups → Quotient Groups → First Isomorphism Theorem for Groups → First Isomorphism Theorem for Rings → First Isomorphism Theorem for Groups → Third Isomorphism Theorem for Groups → First Isomorphism Theorem for Rings → Integral Domains → Principal Ideal Domains → Unique Factorization Domains → Polynomial Rings → Field Extensions → Algebraic and Transcendental Elements → Splitting Fields → Finite Fields → Galois Groups → Fundamental Theorem of Galois Theory → Second Isomorphism Theorem for Groups

Longest path: 99 steps · 411 total prerequisite topics

Prerequisites (4)

First Isomorphism Theorem for Groupshard Third Isomorphism Theorem for Groupssoft Second and Third Isomorphism Theoremssoft Fundamental Theorem of Galois Theorysoft

Leads To (1)

Direct Products of Groupshard