Math 2.0

Group theory and generalizations Theorem

Universal property of the abelianisation

Builds on AbelianisationUniversal property of the quotient groupA quotient is abelian iff the subgroup contains the commutator subgroup

Statement

Let GG be a group, let π ⁣:G→Gab\pi \colon G \to G^{\mathrm{ab}} be the projection to its abelianisation, and let AA be an abelian group. For each group homomorphism φ ⁣:G→A\varphi \colon G \to A, there is exactly one group homomorphism φˉ ⁣:Gab→A\bar{\varphi} \colon G^{\mathrm{ab}} \to A with φˉ∘π=φ\bar{\varphi} \circ \pi = \varphi.

Remarks

Each commutator goes to ee under φ\varphi, because AA is abelian, so [G,G]⊆ker⁡φ[G, G] \subseteq \ker \varphi. Then the universal property of the quotient group gives φˉ\bar{\varphi}. Thus the homomorphisms from GG to AA correspond exactly to the homomorphisms from GabG^{\mathrm{ab}} to AA.

Reading path

To understand this page from the ground up, read these 20 pages in order. The foundations come first.

Show the path as a graph
Group, concept, no evidence levelGroupconceptCommutator of two elements, definition, no evidence levelCommutator of two ele…definitionConjugate elements, definition, no evidence levelConjugate elementsdefinitionGroup homomorphism, definition, no evidence levelGroup homomorphismdefinitionSubgroup, definition, no evidence levelSubgroupdefinitionConjugate subgroup, definition, no evidence levelConjugate subgroupdefinitionKernel of a homomorphism, definition, no evidence levelKernel of a homomorph…definitionLeft 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…definitionCommutator subgroup, definition, no evidence levelCommutator subgroupdefinitionQuotient group, definition, no evidence levelQuotient groupdefinitionA quotient is abelian iff the subgroup contains the commutator subgroup, statement, no evidence levelA quotient is abelian…statementAbelianisation, definition, no evidence levelAbelianisationdefinitionUniversal property of the quotient group, statement, no evidence levelUniversal property of…statementUniversal property of the abelianisation, statement, no evidence levelUniversal property of…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. Group concept
  2. Commutator of two elements definition
  3. Conjugate elements definition
  4. Group homomorphism definition
  5. Subgroup definition
  6. Conjugate subgroup definition
  7. Kernel of a homomorphism definition
  8. Left coset definition
  9. Right coset definition
  10. Subgroup criterion statement
  11. Normal subgroup definition
  12. Intersection of subgroups is a subgroup statement
  13. Normality via left and right cosets statement
  14. Subgroup generated by a subset definition
  15. Commutator subgroup definition
  16. Quotient group definition
  17. A quotient is abelian iff the subgroup contains the commutator subgroup statement
  18. Abelianisation definition
  19. Universal property of the quotient group statement
  20. Universal property of the abelianisation statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Commutator_subgroup, Abelianization
About this page
Tag0EJ4PC8NJ6
URIm2://0EJ4PC8NJ6
Kindstatement, theorem
WritingW0 stub
MSC20F12
StatusThis copy of the site has no build data. Run m2 status 0EJ4PC8NJ6.
UsesAbelianisation, Universal property of the quotient group, A quotient is abelian iff the subgroup contains the commutator subgroup

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