Math 2.0

Group theory and generalizations Foundations Theorem

Third isomorphism theorem

Builds on First isomorphism theoremQuotient group

Statement

Let GG be a group, and let NN and KK be normal subgroups of GG with N⊆KN \subseteq K. Then K/NK/N is a normal subgroup of G/NG/N, and the map (gN)(K/N)↦gK(gN)(K/N) \mapsto gK is an isomorphism from (G/N)/(K/N)(G/N)/(K/N) to G/KG/K.

Remarks

The proof applies the first isomorphism theorem to the surjective homomorphism G/N→G/KG/N \to G/K, gN↦gKgN \mapsto gK, whose kernel is K/NK/N. For example, (ℤ/12ℤ)/(4ℤ/12ℤ)≅ℤ/4ℤ(\mathbb{Z}/12\mathbb{Z})/(4\mathbb{Z}/12\mathbb{Z}) \cong \mathbb{Z}/4\mathbb{Z}. For finite groups, it gives [G:K]=[G/N:K/N][G : K] = [G/N : K/N].

Reading path

To understand this page from the ground up, read these 29 pages in order. The foundations come first. The path takes the proofs Proof that the kernel is a normal subgroup, Proof that a homomorphism is injective iff its kernel is trivial, Proof of the first isomorphism theorem, Proof of the uniqueness of inverses, Proof of the cancellation laws.

Show the path as a graph
Binary operation, definition, no evidence levelBinary operationdefinitionGroup, concept, no evidence levelGroupconceptIdentity element, definition, no evidence levelIdentity elementdefinitionConjugate elements, definition, no evidence levelConjugate elementsdefinitionGroup homomorphism, definition, no evidence levelGroup homomorphismdefinitionSubgroup, definition, no evidence levelSubgroupdefinitionInverse element, definition, no evidence levelInverse elementdefinitionConjugate subgroup, definition, no evidence levelConjugate subgroupdefinitionGroup isomorphism, definition, no evidence levelGroup isomorphismdefinitionKernel of a homomorphism, definition, no evidence levelKernel of a homomorph…definitionImage of a homomorphism, definition, no evidence levelImage of a homomorphi…definitionLeft coset, definition, no evidence levelLeft cosetdefinitionRight coset, definition, no evidence levelRight cosetdefinitionProof of the cancellation laws, proof, no evidence levelProof of the cancella…proofProof of the uniqueness of inverses, proof, no evidence levelProof of the uniquene…proofNormal subgroup, definition, no evidence levelNormal subgroupdefinitionCancellation laws, statement, no evidence levelCancellation lawsstatementUniqueness of inverses, statement, no evidence levelUniqueness of inversesstatementNormality via left and right cosets, statement, no evidence levelNormality via left an…statementHomomorphisms preserve identity and inverses, statement, no evidence levelHomomorphisms preserv…statementQuotient group, definition, no evidence levelQuotient groupdefinitionProof that a homomorphism is injective iff its kernel is trivial, proof, no evidence levelProof that a homomorp…proofProof that the kernel is a normal subgroup, proof, no evidence levelProof that the kernel…proofUniversal property of the quotient group, statement, no evidence levelUniversal property of…statementA homomorphism is injective iff its kernel is trivial, statement, no evidence levelA homomorphism is inj…statementThe kernel is a normal subgroup, statement, no evidence levelThe kernel is a norma…statementProof of the first isomorphism theorem, proof, no evidence levelProof of the first is…proofFirst isomorphism theorem, statement, no evidence levelFirst isomorphism the…statementThird isomorphism theorem, statement, no evidence levelThird isomorphism the…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Binary operation definition
  2. Group concept
  3. Identity element definition
  4. Conjugate elements definition
  5. Group homomorphism definition
  6. Subgroup definition
  7. Inverse element definition
  8. Conjugate subgroup definition
  9. Group isomorphism definition
  10. Kernel of a homomorphism definition
  11. Image of a homomorphism definition
  12. Left coset definition
  13. Right coset definition
  14. Proof of the cancellation laws proof
  15. Proof of the uniqueness of inverses proof
  16. Normal subgroup definition
  17. Cancellation laws statement
  18. Uniqueness of inverses statement
  19. Normality via left and right cosets statement
  20. Homomorphisms preserve identity and inverses statement
  21. Quotient group definition
  22. Proof that a homomorphism is injective iff its kernel is trivial proof
  23. Proof that the kernel is a normal subgroup proof
  24. Universal property of the quotient group statement
  25. A homomorphism is injective iff its kernel is trivial statement
  26. The kernel is a normal subgroup statement
  27. Proof of the first isomorphism theorem proof
  28. First isomorphism theorem statement
  29. Third isomorphism theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Isomorphism_theorems, Theorem C (groups)
About this page
Tag0TJ1V9DDPC
URIm2://0TJ1V9DDPC
Kindstatement, theorem
WritingW0 stub
MSC20A05
StatusThis copy of the site has no build data. Run m2 status 0TJ1V9DDPC.
UsesFirst isomorphism theorem, Quotient group

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