Math 2.0

statement proposition

Order of the alternating group

Statement

Let n≥2n \geq 2. The alternating group AnA_n has order n!/2n!/2, and [Sn:An]=2[S_n : A_n] = 2.

Remarks

The sign map is a surjective homomorphism onto a group of order 22 with kernel AnA_n, so Sn/AnS_n / A_n has order 22. Another proof: the map σ↦(1 2)σ\sigma \mapsto (1\ 2) \sigma is a bijection from the even permutations to the odd permutations. For example, ∣A4∣=12|A_4| = 12 and ∣A5∣=60|A_5| = 60.

Explanations

L1 human

Half of all permutations are even. To see this, multiply by one fixed swap: this changes each even permutation into an odd one, and back. So SnS_n splits into two halves of the same size.

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What this needs

To read this node, read these 35 nodes in this order. The foundations come first, and the list ends with this node.

Binary operation, definition, no evidence levelBinary operationdefinitionCycle notation, concept, no evidence levelCycle notationconceptGroup, concept, no evidence levelGroupconceptIdentity element, definition, no evidence levelIdentity elementdefinitionCancellation laws, statement, no evidence levelCancellation lawsstatementConjugate elements, definition, no evidence levelConjugate elementsdefinitionGroup homomorphism, definition, no evidence levelGroup homomorphismdefinitionOrder of a group, definition, no evidence levelOrder of a groupdefinitionSubgroup, definition, no evidence levelSubgroupdefinitionSymmetric group, definition, no evidence levelSymmetric groupdefinitionInverse element, definition, no evidence levelInverse elementdefinitionGroup isomorphism, definition, no evidence levelGroup isomorphismdefinitionKernel of a homomorphism, definition, no evidence levelKernel of a homomorph…definitionConjugate subgroup, definition, no evidence levelConjugate subgroupdefinitionDisjoint cycles commute, statement, no evidence levelDisjoint cycles commu…statementTransposition, definition, no evidence levelTranspositiondefinitionImage of a homomorphism, definition, no evidence levelImage of a homomorphi…definitionLeft coset, definition, no evidence levelLeft cosetdefinitionOrder of the symmetric group, statement, no evidence levelOrder of the symmetri…statementRight coset, definition, no evidence levelRight cosetdefinitionUniqueness of inverses, statement, no evidence levelUniqueness of inversesstatementDecomposition of a permutation into disjoint cycles, statement, no evidence levelDecomposition of a pe…statementNormal subgroup, definition, no evidence levelNormal subgroupdefinitionHomomorphisms preserve identity and inverses, statement, no evidence levelHomomorphisms preserv…statementDecomposition of a permutation into transpositions, statement, no evidence levelDecomposition of a pe…statementNormality via left and right cosets, statement, no evidence levelNormality via left an…statementSign of a permutation, definition, no evidence levelSign of a permutationdefinitionThe kernel is a normal subgroup, statement, no evidence levelThe kernel is a norma…statementQuotient group, definition, no evidence levelQuotient groupdefinitionEven and odd permutations, definition, no evidence levelEven and odd permutat…definitionSign is a homomorphism, statement, no evidence levelSign is a homomorphismstatementUniversal property of the quotient group, statement, no evidence levelUniversal property of…statementAlternating group, definition, no evidence levelAlternating groupdefinitionFirst isomorphism theorem, statement, no evidence levelFirst isomorphism the…statementOrder of the alternating group, statement, no evidence levelOrder of the alternat…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.
  • no evidence level
  1. Binary operation definition
  2. Cycle notation concept
  3. Group concept
  4. Identity element definition
  5. Cancellation laws statement
  6. Conjugate elements definition
  7. Group homomorphism definition
  8. Order of a group definition
  9. Subgroup definition
  10. Symmetric group definition
  11. Inverse element definition
  12. Group isomorphism definition
  13. Kernel of a homomorphism definition
  14. Conjugate subgroup definition
  15. Disjoint cycles commute statement
  16. Transposition definition
  17. Image of a homomorphism definition
  18. Left coset definition
  19. Order of the symmetric group statement
  20. Right coset definition
  21. Uniqueness of inverses statement
  22. Decomposition of a permutation into disjoint cycles statement
  23. Normal subgroup definition
  24. Homomorphisms preserve identity and inverses statement
  25. Decomposition of a permutation into transpositions statement
  26. Normality via left and right cosets statement
  27. Sign of a permutation definition
  28. The kernel is a normal subgroup statement
  29. Quotient group definition
  30. Even and odd permutations definition
  31. Sign is a homomorphism statement
  32. Universal property of the quotient group statement
  33. Alternating group definition
  34. First isomorphism theorem statement
  35. Order of the alternating group statement

Sources

  1. Web https://en.wikipedia.org/wiki/Alternating_group, Basic properties
  2. Web https://en.wikipedia.org/wiki/Parity_of_a_permutation, Properties

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