Group Isomorphism Theorems and Correspondence
Handbook guide to group isomorphism theorems and correspondence with core definitions, structural results, reasoning methods and verification checks.
How the topic fits together
The Isomorphism Theorems
This source section supplies the definitions, constructions and formal results used in this article. The sequence of results is preserved so later claims are not detached from their prerequisites.
Core definitions and structural results
The following cards retain the mathematical content of the source while condensing long proofs into verification strategies. Numerical examples are treated as examples, not universal requirements.
Factor Theorem Any homomorphism f whose kernel K contains N can be factored
Factor Theorem Any homomorphism f whose kernel K contains N can be factored through G/N. Equivalently, in Figure 1.4.1 there is a unique homomorphism f : G/N → H such that f ◦π = f. Furthermore, (i) f is an epimorphism if and only if f is an epimorphism; (ii) f is a monomorphism if and only if K = N; (iii) f is an isomorphism if and only if f is an epimorphism and K = N.
Proof / verification strategy: Construct the natural group homomorphism, identify its kernel and image, and then use quotient structure to prove the claimed isomorphism or correspondence.
First Isomorphism Theorem If f : G →H is a homomorphism with kernel K,
First Isomorphism Theorem If f : G →H is a homomorphism with kernel K, then the image of f is isomorphic to G/K.
Proof / verification strategy: Construct the natural group homomorphism, identify its kernel and image, and then use quotient structure to prove the claimed isomorphism or correspondence.
Lemma
Let H and N be subgroups of G, with N normal in G. Then (i) HN = NH, and therefore by (1.3.6), HN is a subgroup of G. (ii) N is a normal subgroup of HN. (iii) H ∩N is a normal subgroup of H.
Proof / verification strategy: Work directly from closure, inverses, subgroup tests and the defining action or presentation. Use a small cyclic or permutation model to verify the direction of the argument.
Second Isomorphism Theorem If H and N are subgroups of G, with N normal
Second Isomorphism Theorem If H and N are subgroups of G, with N normal in G, then H/(H ∩N) ∼= HN/N. Note that we write HN/N rather than H/N, since N need not be a subgroup of H.
Proof / verification strategy: Construct the natural group homomorphism, identify its kernel and image, and then use quotient structure to prove the claimed isomorphism or correspondence.
Third Isomorphism Theorem If N and H are normal subgroups of G, with N
Third Isomorphism Theorem If N and H are normal subgroups of G, with N contained in H, then G/H ∼= (G/N)/(H/N), a “cancellation law”.
Proof / verification strategy: Construct the natural group homomorphism, identify its kernel and image, and then use quotient structure to prove the claimed isomorphism or correspondence.
Correspondence Theorem If N is a normal subgroup of G, then the map ψ :
Correspondence Theorem If N is a normal subgroup of G, then the map ψ : H →H/N sets up a one-to-one correspondence between subgroups of G containing N and subgroups of G/N. The inverse of ψ is the map τ : Q →π−1(Q), where π is the canonical epimorphism of G onto G/N. Furthermore, (i) H1 ≤H2 if and only if H1/N ≤H2/N, and in this case, [H2 : H1] = [H2/N : H1/N] (ii) H is a normal subgroup of G if and only if H/N is a normal subgroup of G/N. More generally, (iii) H1 is a normal subgroup of H2 if and only if H1/N is a normal subgroup of H2/N, and in this case, H2/H1 ∼= (H2/N)/H1/N).
Proof / verification strategy: Work directly from closure, inverses, subgroup tests and the defining action or presentation. Use a small cyclic or permutation model to verify the direction of the argument.
Quick-reference relationships
Problem-solving workflow
Identify the ambient group
State the operation, identity, inverses and whether commutativity is available.
Locate the relevant subgroup structure
Check generated subgroups, normality, cosets, stabilisers or direct factors before applying a theorem.
Use the correct counting or mapping tool
Choose coset counting, orbit counting, a homomorphism, a quotient or a group action as appropriate.
Check hypotheses explicitly
Finite-order assumptions, normality, prime-power divisibility and coprimality conditions are not interchangeable.
Translate the result back to structure
Interpret the result as a statement about subgroup size, quotient structure, action, decomposition or presentation.
Verify with a small model
Use a cyclic, symmetric, dihedral or modular example to test the logic without treating the example as the theorem.
Worked-solution emphasis from the supplied source
The supplied worked solutions for this section repeatedly test order, subgroup, coset, kernel, homomorphism, automorphism, injective. These checks are used here as verification themes rather than copied as answer text.
Common mistakes and boundary conditions
- Treating left and right cosets as identical without normality.
- Assuming the converse of a subgroup-order divisibility result.
- Confusing the order of a group with the order of one of its elements.
- Using quotient multiplication before checking that the subgroup is normal.
Verification checklist
- State the ambient algebraic structure and operation before applying a theorem.
- Record every hypothesis that controls the result: finiteness, commutativity, normality, primality, separability, exactness or other section-specific conditions.
- Distinguish a definition from a theorem that follows from it.
- Check whether a map is well-defined before using its kernel, image, inverse or induced map.
- Use a concrete example only as a check; do not promote an illustrative value or pattern to a universal rule.
- When a quotient, localisation or extension is constructed, identify the canonical map and what becomes equal, invertible or fixed.
Source coverage map
| Source section | Subject | PDF pages analysed |
|---|---|---|
| 1.4 | The Isomorphism Theorems | 20–22 |
Related Mathematics pages
Source note: synthesised from the supplied abstract-algebra PDF. The complete 298-page file, including diagrams and worked solutions, was reviewed. Source-identifying author and bibliographic personal details are intentionally omitted. Formal proofs are condensed; the page does not claim requirements or values not supported by the supplied mathematics.