Math 2.0

Group theory and generalizations Definition

Nilpotent group

Builds on Centre of a groupNormal subgroupQuotient group

Definition

A group GG is nilpotent when it has a central series. This is a finite sequence {e}=N0⊆N1⊆⋯⊆Nn=G\{e\} = N_0 \subseteq N_1 \subseteq \cdots \subseteq N_n = G of normal subgroups of GG with Ni+1/Ni⊆Z(G/Ni)N_{i+1}/N_i \subseteq Z(G/N_i) for each i<ni < n. Here Z(G/Ni)Z(G/N_i) is the centre of the quotient group G/NiG/N_i.

Remarks

The condition Ni+1/Ni⊆Z(G/Ni)N_{i+1}/N_i \subseteq Z(G/N_i) means [Ni+1,G]⊆Ni[N_{i+1}, G] \subseteq N_i. Equivalently, the lower central series reaches {e}\{e\}, or the upper central series reaches GG; the least such length is the nilpotency class. The nilpotent groups of class at most 11 are exactly the abelian groups. The quaternion group Q8Q_8 is nilpotent of class 22, but S3S_3 is not nilpotent.

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Group, concept, no evidence levelGroupconceptCentre of a group, definition, no evidence levelCentre of a groupdefinitionConjugate 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 cosetdefinitionNormal subgroup, definition, no evidence levelNormal subgroupdefinitionNormality via left and right cosets, statement, no evidence levelNormality via left an…statementQuotient group, definition, no evidence levelQuotient groupdefinitionNilpotent group, definition, no evidence levelNilpotent groupdefinition
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.
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  1. Group concept
  2. Centre of a group definition
  3. Conjugate elements definition
  4. Subgroup definition
  5. Conjugate subgroup definition
  6. Left coset definition
  7. Right coset definition
  8. Normal subgroup definition
  9. Normality via left and right cosets statement
  10. Quotient group definition
  11. Nilpotent group definition

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Further reading

  1. Web https://en.wikipedia.org/wiki/Nilpotent_group, Definition
  2. Web https://en.wikipedia.org/wiki/Central_series, Definition
About this page
Tag0DZPD2MG33
URIm2://0DZPD2MG33
Kinddefinition
WritingW1 stated
MSC20F18
DefinesNilpotent group
UsesCentre of a group, Normal subgroup, Quotient group

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