Math 2.0

Group theory and generalizations Proposition

Order of the alternating group

Builds on Alternating groupSign is a homomorphismFirst isomorphism theoremOrder of the symmetric 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.

Reading path

To understand this page from the ground up, read these 41 pages in order. The foundations come first. The path takes the proofs Proof that the kernel is a normal subgroup, Proof that a homomorphism is injective iff its kernel is trivial, Proof of the first isomorphism theorem, Proof of the uniqueness of inverses, Proof of the cancellation laws.

  1. Binary operation definition
  2. Cycle notation concept
  3. Group concept
  4. Identity element definition
  5. Conjugate elements definition
  6. Group homomorphism definition
  7. Order of a group definition
  8. Subgroup definition
  9. Symmetric group definition
  10. Inverse element definition
  11. Group isomorphism definition
  12. Kernel of a homomorphism definition
  13. Conjugate subgroup definition
  14. Disjoint cycles commute statement
  15. Image of a homomorphism definition
  16. Transposition definition
  17. Left coset definition
  18. Order of the symmetric group statement
  19. Right coset definition
  20. Proof of the cancellation laws proof
  21. Proof of the uniqueness of inverses proof
  22. Decomposition of a permutation into disjoint cycles statement
  23. Normal subgroup definition
  24. Cancellation laws statement
  25. Uniqueness of inverses statement
  26. Decomposition of a permutation into transpositions statement
  27. Normality via left and right cosets statement
  28. Homomorphisms preserve identity and inverses statement
  29. Sign of a permutation definition
  30. Quotient group definition
  31. Even and odd permutations definition
  32. Proof that a homomorphism is injective iff its kernel is trivial proof
  33. Sign is a homomorphism statement
  34. Proof that the kernel is a normal subgroup proof
  35. Universal property of the quotient group statement
  36. A homomorphism is injective iff its kernel is trivial statement
  37. Alternating group definition
  38. The kernel is a normal subgroup statement
  39. Proof of the first isomorphism theorem proof
  40. First isomorphism theorem statement
  41. Order of the alternating group statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Alternating_group, Basic properties
  2. Web https://en.wikipedia.org/wiki/Parity_of_a_permutation, Properties
About this page
Tag0SZ1KMNWVE
URIm2://0SZ1KMNWVE
Kindstatement, proposition
WritingW0 stub
MSC20B30
StatusThis copy of the site has no build data. Run m2 status 0SZ1KMNWVE.
UsesAlternating group, Sign is a homomorphism, First isomorphism theorem, Order of the symmetric group

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

No page links to this page yet.