Math 2.0

Group theory and generalizations Theorem

The symmetric group on five or more letters is not solvable

Builds on Simplicity of the alternating groupSubgroups of solvable groups are solvableDerived series

Statement

Let n≥5n \geq 5. Then the symmetric group SnS_n is not solvable.

Remarks

The group AnA_n is simple and not abelian, so its commutator subgroup is AnA_n, and the derived series of AnA_n is constant. So AnA_n is not solvable, and then SnS_n is not solvable, because subgroups of solvable groups are solvable. The groups S1,S2,S3S_1, S_2, S_3 and S4S_4 are solvable. By Galois theory, this is the reason why the general polynomial equation of degree n≥5n \geq 5 is not solvable by radicals.

Reading path

To understand this page from the ground up, read these 37 pages in order. The foundations come first.

Show the path as a graph
Binary operation, definition, no evidence levelBinary operationdefinitionCycle notation, concept, no evidence levelCycle notationconceptGroup, concept, no evidence levelGroupconceptCommutativity, definition, no evidence levelCommutativitydefinitionCommutator of two elements, definition, no evidence levelCommutator of two ele…definitionConjugate elements, definition, no evidence levelConjugate elementsdefinitionGroup homomorphism, definition, no evidence levelGroup homomorphismdefinitionSubgroup, definition, no evidence levelSubgroupdefinitionSymmetric group, definition, no evidence levelSymmetric groupdefinitionAbelian group, definition, no evidence levelAbelian groupdefinitionConjugate subgroup, definition, no evidence levelConjugate subgroupdefinitionDisjoint cycles commute, statement, no evidence levelDisjoint cycles commu…statementKernel of a homomorphism, definition, no evidence levelKernel of a homomorph…definitionTransposition, definition, no evidence levelTranspositiondefinitionLeft coset, definition, no evidence levelLeft cosetdefinitionRight coset, definition, no evidence levelRight cosetdefinitionSubgroup criterion, statement, no evidence levelSubgroup criterionstatementNormal subgroup, definition, no evidence levelNormal subgroupdefinitionDecomposition of a permutation into disjoint cycles, statement, no evidence levelDecomposition of a pe…statementIntersection of subgroups is a subgroup, statement, no evidence levelIntersection of subgr…statementDecomposition of a permutation into transpositions, statement, no evidence levelDecomposition of a pe…statementSubgroup generated by a subset, definition, no evidence levelSubgroup generated by…definitionNormality via left and right cosets, statement, no evidence levelNormality via left an…statementSign of a permutation, definition, no evidence levelSign of a permutationdefinitionCommutator subgroup, definition, no evidence levelCommutator subgroupdefinitionQuotient group, definition, no evidence levelQuotient groupdefinitionEven and odd permutations, definition, no evidence levelEven and odd permutat…definitionSign of a cycle, statement, no evidence levelSign of a cyclestatementDerived series, definition, no evidence levelDerived seriesdefinitionSubnormal series, definition, no evidence levelSubnormal seriesdefinitionSign is a homomorphism, statement, no evidence levelSign is a homomorphismstatementSolvable group, definition, no evidence levelSolvable groupdefinitionAlternating group, definition, no evidence levelAlternating groupdefinitionThe alternating group is generated by three-cycles, statement, no evidence levelThe alternating group…statementSubgroups of solvable groups are solvable, statement, no evidence levelSubgroups of solvable…statementSimplicity of the alternating group, statement, no evidence levelSimplicity of the alt…statementThe symmetric group on five or more letters is not solvable, statement, no evidence levelThe symmetric group o…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. Cycle notation concept
  3. Group concept
  4. Commutativity definition
  5. Commutator of two elements definition
  6. Conjugate elements definition
  7. Group homomorphism definition
  8. Subgroup definition
  9. Symmetric group definition
  10. Abelian group definition
  11. Conjugate subgroup definition
  12. Disjoint cycles commute statement
  13. Kernel of a homomorphism definition
  14. Transposition definition
  15. Left coset definition
  16. Right coset definition
  17. Subgroup criterion statement
  18. Normal subgroup definition
  19. Decomposition of a permutation into disjoint cycles statement
  20. Intersection of subgroups is a subgroup statement
  21. Decomposition of a permutation into transpositions statement
  22. Subgroup generated by a subset definition
  23. Normality via left and right cosets statement
  24. Sign of a permutation definition
  25. Commutator subgroup definition
  26. Quotient group definition
  27. Even and odd permutations definition
  28. Sign of a cycle statement
  29. Derived series definition
  30. Subnormal series definition
  31. Sign is a homomorphism statement
  32. Solvable group definition
  33. Alternating group definition
  34. The alternating group is generated by three-cycles statement
  35. Subgroups of solvable groups are solvable statement
  36. Simplicity of the alternating group statement
  37. The symmetric group on five or more letters is not solvable statement

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Solvable_group, Non-example
About this page
Tag0J53NE674D
URIm2://0J53NE674D
Kindstatement, theorem
WritingW0 stub
MSC20F16, 20B30
StatusThis copy of the site has no build data. Run m2 status 0J53NE674D.
UsesSimplicity of the alternating group, Subgroups of solvable groups are solvable, Derived series

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

Used by