Math 2.0

Group theory and generalizations Group actions and permutation groups Theorem

Simplicity of the alternating group

Builds on Alternating groupThe alternating group is generated by three-cyclesNormal subgroup

Statement

Let n≥5n \geq 5. Then the alternating group AnA_n is simple: its only normal subgroups are {e}\{e\} and AnA_n.

Remarks

A usual proof shows that a nontrivial normal subgroup of AnA_n contains a 33-cycle, then all 33-cycles, and then uses that the 3-cycles generate AnA_n. The group A4A_4 is not simple, because the Klein four-group {e,(1 2)(3 4),(1 3)(2 4),(1 4)(2 3)}\{e, (1\ 2)(3\ 4), (1\ 3)(2\ 4), (1\ 4)(2\ 3)\} is normal in A4A_4. The group A3A_3 is simple because it has prime order 33. For n≥5n \geq 5, the only normal subgroups of SnS_n are {e}\{e\}, AnA_n and SnS_n.

Reading path

To understand this page from the ground up, read these 23 pages in order. The foundations come first.

Show the path as a graph
Cycle notation, concept, no evidence levelCycle notationconceptGroup, concept, no evidence levelGroupconceptConjugate elements, definition, no evidence levelConjugate elementsdefinitionGroup homomorphism, definition, no evidence levelGroup homomorphismdefinitionSubgroup, definition, no evidence levelSubgroupdefinitionSymmetric group, definition, no evidence levelSymmetric groupdefinitionConjugate subgroup, definition, no evidence levelConjugate subgroupdefinitionKernel of a homomorphism, definition, no evidence levelKernel of a homomorph…definitionDisjoint cycles commute, statement, no evidence levelDisjoint cycles commu…statementTransposition, definition, no evidence levelTranspositiondefinitionSubgroup 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…definitionSign of a permutation, definition, no evidence levelSign of a permutationdefinitionEven and odd permutations, definition, no evidence levelEven and odd permutat…definitionSign of a cycle, statement, no evidence levelSign of a cyclestatementSign is a homomorphism, statement, no evidence levelSign is a homomorphismstatementAlternating group, definition, no evidence levelAlternating groupdefinitionThe alternating group is generated by three-cycles, statement, no evidence levelThe alternating group…statementSimplicity of the alternating group, statement, no evidence levelSimplicity of the alt…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. Cycle notation concept
  2. Group concept
  3. Conjugate elements definition
  4. Group homomorphism definition
  5. Subgroup definition
  6. Symmetric group definition
  7. Conjugate subgroup definition
  8. Kernel of a homomorphism definition
  9. Disjoint cycles commute statement
  10. Transposition definition
  11. Subgroup criterion statement
  12. Normal subgroup definition
  13. Decomposition of a permutation into disjoint cycles statement
  14. Intersection of subgroups is a subgroup statement
  15. Decomposition of a permutation into transpositions statement
  16. Subgroup generated by a subset definition
  17. Sign of a permutation definition
  18. Even and odd permutations definition
  19. Sign of a cycle statement
  20. Sign is a homomorphism statement
  21. Alternating group definition
  22. The alternating group is generated by three-cycles statement
  23. Simplicity of the alternating group statement

Where this leads

4 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Alternating_group, Basic properties
  2. Web https://en.wikipedia.org/wiki/Alternating_group, Generators and relations
About this page
Tag0ARPRJQWZ2
URIm2://0ARPRJQWZ2
Kindstatement, theorem
WritingW0 stub
MSC20B30, 20D06
StatusThis copy of the site has no build data. Run m2 status 0ARPRJQWZ2.
UsesAlternating group, The alternating group is generated by three-cycles, Normal subgroup

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