Math 2.0

Group theory and generalizations Foundations Theorem

Correspondence theorem

Builds on Quotient groupNormal subgroupLattice of subgroups

Statement

Let GG be a group and let NN be a normal subgroup of GG. Then the map K↦K/NK \mapsto K/N is a bijection from the set of subgroups KK of GG with N⊆KN \subseteq K to the set of all subgroups of G/NG/N. Here G/NG/N is the quotient group. Let KK and LL be subgroups of GG that contain NN. Then K⊆LK \subseteq L if and only if K/N⊆L/NK/N \subseteq L/N. Also, KK is normal in GG if and only if K/NK/N is normal in G/NG/N.

Remarks

The inverse map sends a subgroup Kˉ\bar{K} of G/NG/N to its preimage {g∈G:gN∈Kˉ}\{g \in G : gN \in \bar{K}\}. The bijection is an isomorphism of lattices of subgroups, and it keeps indices: [L:K]=[L/N:K/N][L : K] = [L/N : K/N] when N⊆K⊆LN \subseteq K \subseteq L. For example, the subgroups of ℤ/12ℤ\mathbb{Z}/12\mathbb{Z} correspond to the subgroups dℤd\mathbb{Z} of ℤ\mathbb{Z} with dd dividing 1212.

Reading path

To understand this page from the ground up, read these 14 pages in order. The foundations come first.

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Group, concept, no evidence levelGroupconceptConjugate elements, definition, no evidence levelConjugate elementsdefinitionSubgroup, definition, no evidence levelSubgroupdefinitionConjugate subgroup, definition, no evidence levelConjugate subgroupdefinitionLeft coset, definition, no evidence levelLeft cosetdefinitionRight coset, definition, no evidence levelRight cosetdefinitionSubgroup criterion, statement, no evidence levelSubgroup criterionstatementNormal subgroup, definition, no evidence levelNormal subgroupdefinitionIntersection of subgroups is a subgroup, statement, no evidence levelIntersection of subgr…statementNormality via left and right cosets, statement, no evidence levelNormality via left an…statementSubgroup generated by a subset, definition, no evidence levelSubgroup generated by…definitionQuotient group, definition, no evidence levelQuotient groupdefinitionLattice of subgroups, concept, no evidence levelLattice of subgroupsconceptCorrespondence theorem, statement, no evidence levelCorrespondence theoremstatement
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. Group concept
  2. Conjugate elements definition
  3. Subgroup definition
  4. Conjugate subgroup definition
  5. Left coset definition
  6. Right coset definition
  7. Subgroup criterion statement
  8. Normal subgroup definition
  9. Intersection of subgroups is a subgroup statement
  10. Normality via left and right cosets statement
  11. Subgroup generated by a subset definition
  12. Quotient group definition
  13. Lattice of subgroups concept
  14. Correspondence theorem statement

Where this leads

3 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Correspondence_theorem, lead section
  2. Web https://en.wikipedia.org/wiki/Isomorphism_theorems, Theorem D (groups)
About this page
Tag0S9ZVZ34MD
URIm2://0S9ZVZ34MD
Aliaseslattice-theorem, fourth-isomorphism-theorem
Kindstatement, theorem
WritingW1 stated
MSC20A05
StatusThis copy of the site has no build data. Run m2 status 0S9ZVZ34MD.
UsesQuotient group, Normal subgroup, Lattice of subgroups
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