Math 2.0

Group theory and generalizations Proposition

Nilpotent groups are solvable

Builds on Nilpotent groupSolvable group

Statement

Let GG be a nilpotent group. Then GG is solvable.

Remarks

A central series {e}=N0⊆⋯⊆Nn=G\{e\} = N_0 \subseteq \cdots \subseteq N_n = G is a subnormal series, and each factor Ni+1/NiN_{i+1}/N_i is central in G/NiG/N_i, so it is abelian. The converse is false: S3S_3 is solvable, but it is not nilpotent. So there are strict inclusions: abelian groups, then nilpotent groups, then solvable groups.

Reading path

To understand this page from the ground up, read these 17 pages in order. The foundations come first.

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Binary operation, definition, no evidence levelBinary operationdefinitionGroup, concept, no evidence levelGroupconceptCommutativity, definition, no evidence levelCommutativitydefinitionCentre of a group, definition, no evidence levelCentre of a groupdefinitionConjugate elements, definition, no evidence levelConjugate elementsdefinitionSubgroup, definition, no evidence levelSubgroupdefinitionAbelian group, definition, no evidence levelAbelian groupdefinitionConjugate 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 groupdefinitionSubnormal series, definition, no evidence levelSubnormal seriesdefinitionNilpotent group, definition, no evidence levelNilpotent groupdefinitionSolvable group, definition, no evidence levelSolvable groupdefinitionNilpotent groups are solvable, statement, no evidence levelNilpotent groups are …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. Binary operation definition
  2. Group concept
  3. Commutativity definition
  4. Centre of a group definition
  5. Conjugate elements definition
  6. Subgroup definition
  7. Abelian group definition
  8. Conjugate subgroup definition
  9. Left coset definition
  10. Right coset definition
  11. Normal subgroup definition
  12. Normality via left and right cosets statement
  13. Quotient group definition
  14. Subnormal series definition
  15. Nilpotent group definition
  16. Solvable group definition
  17. Nilpotent groups are solvable statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Nilpotent_group, Properties
About this page
Tag0GNG2KVPZJ
URIm2://0GNG2KVPZJ
Kindstatement, proposition
WritingW0 stub
MSC20F18, 20F16
StatusThis copy of the site has no build data. Run m2 status 0GNG2KVPZJ.
UsesNilpotent group, Solvable group

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