Math 2.0

Group theory and generalizations Proposition

Abelian groups are solvable

Builds on Solvable groupAbelian group

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.

Reading path

To understand this page from the ground up, read these 15 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 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 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. 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

Further reading

  1. Web https://en.wikipedia.org/wiki/Solvable_group, Abelian groups
About this page
Tag0R0X73XHAT
URIm2://0R0X73XHAT
Kindstatement, proposition
WritingW0 stub
MSC20F16
StatusThis copy of the site has no build data. Run m2 status 0R0X73XHAT.
UsesSolvable group, Abelian group

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