Math 2.0

Group theory and generalizations Definition

Abelianisation

Builds on Commutator subgroupQuotient group

Definition

Let GG be a group and let [G,G][G, G] be its commutator subgroup. The abelianisation of GG is the quotient group Gab=G/[G,G]G^{\mathrm{ab}} = G/[G, G].

Remarks

The group GabG^{\mathrm{ab}} is abelian, by this criterion. If GG is abelian, then Gab=GG^{\mathrm{ab}} = G. The abelianisation of the symmetric group SnS_n for n≥2n \geq 2 is cyclic of order 22, and the sign of a permutation gives the isomorphism.

Reading path

To understand this page from the ground up, read these 15 pages in order. The foundations come first.

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Group, concept, no evidence levelGroupconceptCommutator of two elements, definition, no evidence levelCommutator of two ele…definitionConjugate 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 groupdefinitionCommutator subgroup, definition, no evidence levelCommutator subgroupdefinitionAbelianisation, definition, no evidence levelAbelianisationdefinition
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. Subgroup definition
  5. Conjugate subgroup definition
  6. Left coset definition
  7. Right coset definition
  8. Subgroup criterion statement
  9. Normal subgroup definition
  10. Intersection of subgroups is a subgroup statement
  11. Normality via left and right cosets statement
  12. Subgroup generated by a subset definition
  13. Quotient group definition
  14. Commutator subgroup definition
  15. Abelianisation definition

Where this leads

11 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Commutator_subgroup, Abelianization
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Kinddefinition
WritingW0 stub
MSC20F12
DefinesAbelianisation
UsesCommutator subgroup, Quotient group

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