Math 2.0

statement proposition

Abelian groups are solvable

Statement

Let GG be an abelian group. Then GG is solvable.

Remarks

The series {e}⊴G\{e\} \trianglelefteq G has one factor, GG itself, and this factor is abelian. Equivalently, the commutator subgroup G(1)G^{(1)} is already {e}\{e\}. The converse is false: S3S_3 is solvable, by the series {e}⊴A3⊴S3\{e\} \trianglelefteq A_3 \trianglelefteq S_3, but it is not abelian.

Explanations

L1 human

An abelian group is already one abelian layer. So it is solvable with a series of length one. Solvable groups generalise abelian groups.

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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.

Binary operation, definition, no evidence levelBinary operationdefinitionGroup, concept, no evidence levelGroupconceptCommutativity, definition, no evidence levelCommutativitydefinitionConjugate 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 seriesdefinitionSolvable group, definition, no evidence levelSolvable groupdefinitionAbelian groups are solvable, statement, no evidence levelAbelian groups are so…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. Conjugate elements definition
  5. Subgroup definition
  6. Abelian group definition
  7. Conjugate subgroup definition
  8. Left coset definition
  9. Right coset definition
  10. Normal subgroup definition
  11. Normality via left and right cosets statement
  12. Quotient group definition
  13. Subnormal series definition
  14. Solvable group definition
  15. Abelian groups are solvable statement

Sources

  1. Web https://en.wikipedia.org/wiki/Solvable_group, Abelian groups

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