Math 2.0

Group theory and generalizations Foundations Theorem

Second isomorphism theorem

Builds on First isomorphism theoremProduct of a subgroup and a normal subgroup is a subgroupQuotient group

Statement

Let GG be a group, let SS be a subgroup of GG, and let NN be a normal subgroup of GG. Then SNSN is a subgroup of GG, NN is normal in SNSN, and S∩NS \cap N is normal in SS. Moreover, the map s(S∩N)↦sNs(S \cap N) \mapsto sN is an isomorphism from S/(S∩N)S/(S \cap N) to SN/NSN/N.

Remarks

The proof applies the first isomorphism theorem to the homomorphism S→SN/NS \to SN/N, s↦sNs \mapsto sN, which is surjective with kernel S∩NS \cap N. For finite groups, it gives ∣SN∣=∣S∣ ∣N∣/∣S∩N∣|SN| = |S| \, |N| / |S \cap N|. It is enough that SS normalizes NN, that is, sNs−1=NsNs^{-1} = N for each s∈Ss \in S.

Reading path

To understand this page from the ground up, read these 31 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 cosetdefinitionProduct of two subgroups, definition, no evidence levelProduct of two subgro…definitionRight 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 inversesstatementProduct of a subgroup and a normal subgroup is a subgroup, statement, no evidence levelProduct of a subgroup…statementNormality 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…statementSecond isomorphism theorem, statement, no evidence levelSecond isomorphism th…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. Product of two subgroups definition
  14. Right coset definition
  15. Proof of the cancellation laws proof
  16. Proof of the uniqueness of inverses proof
  17. Normal subgroup definition
  18. Cancellation laws statement
  19. Uniqueness of inverses statement
  20. Product of a subgroup and a normal subgroup is a subgroup statement
  21. Normality via left and right cosets statement
  22. Homomorphisms preserve identity and inverses statement
  23. Quotient group definition
  24. Proof that a homomorphism is injective iff its kernel is trivial proof
  25. Proof that the kernel is a normal subgroup proof
  26. Universal property of the quotient group statement
  27. A homomorphism is injective iff its kernel is trivial statement
  28. The kernel is a normal subgroup statement
  29. Proof of the first isomorphism theorem proof
  30. First isomorphism theorem statement
  31. Second isomorphism theorem statement

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Isomorphism_theorems, Theorem B (groups)
About this page
Tag0PPP2J2DJV
URIm2://0PPP2J2DJV
Aliasesdiamond-isomorphism-theorem
Kindstatement, theorem
WritingW0 stub
MSC20A05
StatusThis copy of the site has no build data. Run m2 status 0PPP2J2DJV.
UsesFirst isomorphism theorem, Product of a subgroup and a normal subgroup is a subgroup, Quotient group

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