Math 2.0

statement proposition

Nilpotent groups are solvable

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.

Explanations

L1 human

A central series has layers that commute with the whole group. In particular, each layer is abelian. So a nilpotent group is solvable.

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To read this node, read these 17 nodes in this order. The foundations come first, and the list ends with this node.

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 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.
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  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

Sources

  1. Web https://en.wikipedia.org/wiki/Nilpotent_group, Properties

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