Math 2.0

statement proposition

A quotient is abelian iff the subgroup contains the commutator 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.

Explanations

L1 human

To make a group abelian by a quotient, you must send each commutator to the identity. So the subgroup that you divide by must contain all commutators. If it does, the quotient is abelian.

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To read this node, read these 15 nodes in this order. The foundations come first, and the list ends with this node.

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 node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.
  • 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

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Sources

  1. Web https://en.wikipedia.org/wiki/Commutator_subgroup, Abelianization

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