Math 2.0

Group theory and generalizations Proof

Proof of the order of the alternating group

Builds on Alternating groupEven and odd permutationsSign is a homomorphismThe kernel is a normal subgroupOrder of the symmetric groupOrder of a group is the index times the order of the subgroupTranspositionSign of a permutation

Proof

We pair the even and the odd permutations by a bijection. Let n≥2n \geq 2.

  1. By the definition of the alternating group, AnA_n is the set of even permutations, that is, of σ∈Sn\sigma \in S_n with sgn⁡(σ)=1\operatorname{sgn}(\sigma) = 1. Let OO be the set of odd permutations, with sign −1-1.
  2. Each permutation has sign 11 or −1-1, so SnS_n is the disjoint union of AnA_n and OO. Hence ∣Sn∣=∣An∣+∣O∣|S_n| = |A_n| + |O|.
  3. The sign is a homomorphism, and AnA_n is its kernel. So AnA_n is a subgroup of SnS_n.
  4. Let t=(1 2)t = (1\ 2). It is a transposition, so sgn⁡(t)=−1\operatorname{sgn}(t) = -1 by the definition of the sign.
  5. Define L ⁣:Sn→SnL \colon S_n \to S_n by L(σ)=tσL(\sigma) = t\sigma. Since the sign is a homomorphism, sgn⁡(L(σ))=−sgn⁡(σ)\operatorname{sgn}(L(\sigma)) = -\operatorname{sgn}(\sigma).
  6. By step 5, LL maps AnA_n into OO and maps OO into AnA_n.
  7. Since tt=et t = e, we get L(L(σ))=σL(L(\sigma)) = \sigma for all σ\sigma. So the restriction An→OA_n \to O of LL has the inverse O→AnO \to A_n, also given by LL.
  8. Thus LL gives a bijection from AnA_n to OO, and ∣An∣=∣O∣|A_n| = |O|.
  9. By the order of the symmetric group, ∣Sn∣=n!|S_n| = n!. With steps 2 and 8, n!=2∣An∣n! = 2|A_n|, so ∣An∣=n!/2|A_n| = n!/2.
  10. Since AnA_n is a subgroup of the finite group SnS_n, we have [[∣Sn∣=[Sn:An]⋅∣An∣|S_n| = [S_n : A_n] \cdot |A_n||04DC0SRN2K]]. So [Sn:An]=n!/(n!/2)=2[S_n : A_n] = n! / (n!/2) = 2.

Idea

Multiplying by one fixed transposition flips the sign of every permutation. It is its own inverse, so it matches the even permutations one to one with the odd ones. Hence exactly half of the n!n! permutations are even.

Reading path

To understand this page from the ground up, read these 126 pages in order. The foundations come first. The path takes the proofs Proof of the decomposition of a permutation into transpositions, Proof of the commutativity of addition by induction, Proof that equivalence classes form a partition, Proof that a composition of injections is injective, Proof that disjoint cycles commute, Proof that a function is bijective iff it has a two-sided inverse, Proof of the order of the symmetric group, Proof that the kernel is a normal subgroup, Proof of strong induction from ordinary induction, Proof of the laws of exponents in a group, Proof of the complement principle, Proof of the number of injections, Proof that a composition of surjections is surjective, Proof that the sign is a homomorphism, Proof that the order of a finite group is the index times the order of a subgroup, Proof of the decomposition of a permutation into disjoint cycles, Proof of the rule of sum, Proof of the well-ordering principle from strong induction, Proof of the division algorithm, Proof of the uniqueness of inverses, Proof of the number of permutations by counting injections, Proof of the bijection principle, Proof that homomorphisms preserve identity and inverses, Proof that cosets have the same cardinality as the subgroup, Proof of the cancellation laws.

  1. Binary operation definition
  2. Cardinality of a finite set definition
  3. Cycle notation concept
  4. Definition by recursion concept
  5. Divisibility definition
  6. Element definition
  7. Factorial definition
  8. Falling factorial definition
  9. Group concept
  10. Mathematical induction concept
  11. Natural numbers object
  12. Peano axioms concept
  13. Permutation definition
  14. Proposition concept
  15. Identity element definition
  16. Set definition
  17. Conjugate elements definition
  18. Group homomorphism definition
  19. Order of a group definition
  20. Subgroup definition
  21. Symmetric group definition
  22. Successor function definition
  23. Conjunction definition
  24. Disjunction definition
  25. Implication definition
  26. Predicate definition
  27. Inverse element definition
  28. Empty set object
  29. Family of sets definition
  30. Ordered pair definition
  31. Kernel of a homomorphism definition
  32. Conjugate subgroup definition
  33. Transposition definition
  34. Intersection definition
  35. Left coset definition
  36. Right coset definition
  37. Union definition
  38. Set-builder notation concept
  39. Addition of natural numbers definition
  40. Biconditional definition
  41. The successor function is injective statement
  42. Zero is not a successor statement
  43. Existential quantifier definition
  44. Universal quantifier definition
  45. Cartesian product definition
  46. Disjoint sets definition
  47. Proof of the cancellation laws proof
  48. Proof of the uniqueness of inverses proof
  49. Normal subgroup definition
  50. Index of a subgroup definition
  51. Subset definition
  52. Axiom of extensionality statement
  53. Proof of the commutativity of addition by induction proof
  54. Union of a family of sets definition
  55. Order on the natural numbers definition
  56. Uniqueness quantifier definition
  57. Binary relation definition
  58. Uniqueness of inverses statement
  59. Cancellation laws statement
  60. Partition of a set definition
  61. Commutativity of addition statement
  62. Proof of strong induction from ordinary induction proof
  63. Reflexive relation definition
  64. Symmetric relation definition
  65. Transitive relation definition
  66. Proof that homomorphisms preserve identity and inverses proof
  67. Proof of the laws of exponents in a group proof
  68. Function definition
  69. Strong induction statement
  70. Equivalence relation definition
  71. Homomorphisms preserve identity and inverses statement
  72. Codomain definition
  73. Function composition definition
  74. Identity function definition
  75. Injective function definition
  76. Restriction of a function definition
  77. Laws of exponents in a group statement
  78. Proof of the well-ordering principle from strong induction proof
  79. Proof that the kernel is a normal subgroup proof
  80. Left inverse definition
  81. Proof that disjoint cycles commute proof
  82. Right inverse definition
  83. Surjective function definition
  84. Proof that a composition of injections is injective proof
  85. Equivalence class definition
  86. Well-ordering principle statement
  87. The kernel is a normal subgroup statement
  88. Disjoint cycles commute statement
  89. Proof that equivalence classes form a partition proof
  90. Proof that a composition of surjections is surjective proof
  91. Composition of injections is injective statement
  92. Bijective function definition
  93. Proof of the division algorithm proof
  94. Proof of the rule of sum proof
  95. Composition of surjections is surjective statement
  96. Equivalence classes form a partition statement
  97. Proof that a function is bijective iff it has a two-sided inverse proof
  98. Division algorithm statement
  99. Equinumerous sets definition
  100. Rule of sum statement
  101. Bijective iff it has a two-sided inverse statement
  102. Proof of the decomposition of a permutation into disjoint cycles proof
  103. Cardinality definition
  104. Proof of the complement principle proof
  105. Proof of the bijection principle proof
  106. Decomposition of a permutation into disjoint cycles statement
  107. Proof that cosets have the same cardinality as the subgroup proof
  108. Complement principle statement
  109. Bijection principle statement
  110. Proof of the decomposition of a permutation into transpositions proof
  111. Cosets have the same cardinality as the subgroup statement
  112. Proof of the number of injections proof
  113. Decomposition of a permutation into transpositions statement
  114. Proof that the order of a finite group is the index times the order of a subgroup proof
  115. Sign of a permutation definition
  116. Number of injections statement
  117. Order of a group is the index times the order of the subgroup statement
  118. Even and odd permutations definition
  119. Proof that the sign is a homomorphism proof
  120. Proof of the number of permutations by counting injections proof
  121. Sign is a homomorphism statement
  122. Number of permutations statement
  123. Alternating group definition
  124. Proof of the order of the symmetric group proof
  125. Order of the symmetric group statement
  126. Proof of the order of the alternating group proof

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Parity_of_a_permutation, Properties
About this page
Tag9T2QXHAPNR
URIm2://9T2QXHAPNR
Kindproof
WritingW0 stub
MSC20B30
ProvesOrder of the alternating group
UsesAlternating group, Even and odd permutations, Sign is a homomorphism, The kernel is a normal subgroup, Order of the symmetric group, Order of a group is the index times the order of the subgroup, Transposition, Sign of a permutation
Changed4 November 2026

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