Math 2.0

Group theory and generalizations Proposition

A quotient is abelian iff the subgroup contains the commutator subgroup

Builds on Commutator subgroupQuotient groupNormal subgroup

Statement

Let GG be a group and let NN be a normal subgroup of GG. Then the quotient group G/NG/N is abelian if and only if [G,G]⊆N[G, G] \subseteq N, where [G,G][G, G] is the commutator subgroup.

Remarks

The cosets gNgN and hNhN commute exactly when g−1h−1gh∈Ng^{-1} h^{-1} g h \in N. Thus [G,G][G, G] is the smallest normal subgroup of GG with an abelian quotient. A subgroup that contains [G,G][G, G] is automatically normal.

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 subgroupdefinitionA quotient is abelian iff the subgroup contains the commutator subgroup, statement, no evidence levelA quotient is abelian…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. 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. A quotient is abelian iff the subgroup contains the commutator subgroup statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Commutator_subgroup, Abelianization
About this page
Tag0MS8QGA6S5
URIm2://0MS8QGA6S5
Kindstatement, proposition
WritingW0 stub
MSC20F12
StatusThis copy of the site has no build data. Run m2 status 0MS8QGA6S5.
UsesCommutator subgroup, Quotient group, Normal subgroup

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