Math 2.0

Group theory and generalizations Definition

Commutator subgroup

Builds on Commutator of two elementsSubgroup generated by a subset

Definition

Let GG be a group. The commutator subgroup of GG is the subgroup generated by the set of all commutators [g,h][g, h] with g,h∈Gg, h \in G. It is written [G,G][G, G] or G′G'.

Remarks

A product of commutators need not be a commutator, so in general the set of commutators is not a subgroup. The subgroup [G,G][G, G] is characteristic, so it is normal in GG. The group GG is abelian if and only if [G,G]={e}[G, G] = \{e\}. The commutator subgroup of the symmetric group SnS_n is the alternating group AnA_n.

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Group, concept, no evidence levelGroupconceptCommutator of two elements, definition, no evidence levelCommutator of two ele…definitionSubgroup, definition, no evidence levelSubgroupdefinitionSubgroup criterion, statement, no evidence levelSubgroup criterionstatementIntersection of subgroups is a subgroup, statement, no evidence levelIntersection of subgr…statementSubgroup generated by a subset, definition, no evidence levelSubgroup generated by…definitionCommutator subgroup, definition, no evidence levelCommutator subgroupdefinition
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  1. Group concept
  2. Commutator of two elements definition
  3. Subgroup definition
  4. Subgroup criterion statement
  5. Intersection of subgroups is a subgroup statement
  6. Subgroup generated by a subset definition
  7. Commutator subgroup definition

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21 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Commutator_subgroup, Definition
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DefinesCommutator subgroup
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