Math 2.0

Group theory and generalizations Proof

Proof that the symmetric group on five or more letters is not solvable

Builds on Solvable groupSubnormal seriesSimple groupSimplicity of the alternating groupAlternating groupEven and odd permutationsSubgroups of solvable groups are solvableSymmetric group

Proof

We argue by contradiction, through the alternating group.

  1. Let n≥5n \geq 5. The alternating group AnA_n is the subgroup of even permutations of the symmetric group SnS_n.
  2. We compose permutations from right to left. Then (1 2 3)=(1 3)(1 2)(1\,2\,3) = (1\,3)(1\,2) and (3 4 5)=(3 5)(3 4)(3\,4\,5) = (3\,5)(3\,4). So both 33-cycles are even and lie in AnA_n; here we use n≥5n \geq 5.
  3. Let a=(1 2 3)a = (1\,2\,3) and b=(3 4 5)b = (3\,4\,5). Then abab sends 33 to a(4)=4a(4) = 4, while baba sends 33 to b(1)=1b(1) = 1. So ab≠baab \neq ba, and AnA_n is not abelian.
  4. By the simplicity of the alternating group, AnA_n is simple.
  5. Suppose that AnA_n is solvable. Take a subnormal series {e}=G0⊴⋯⊴Gk=An\{e\} = G_0 \trianglelefteq \cdots \trianglelefteq G_k = A_n with abelian factors.
  6. Since An≠{e}A_n \neq \{e\}, some GjG_j equals AnA_n with j≥1j \geq 1. Let jj be the least such index. Then Gj−1≠AnG_{j-1} \neq A_n.
  7. The subgroup Gj−1G_{j-1} is normal in Gj=AnG_j = A_n. Since AnA_n is simple and Gj−1≠AnG_{j-1} \neq A_n, we get Gj−1={e}G_{j-1} = \{e\}.
  8. So the factor Gj/Gj−1=An/{e}G_j/G_{j-1} = A_n/\{e\} is abelian. The map x↦x{e}x \mapsto x\{e\} is an isomorphism from AnA_n onto An/{e}A_n/\{e\}, so AnA_n is abelian.
  9. This contradicts step 3. So AnA_n is not solvable.
  10. If SnS_n were solvable, then its subgroup AnA_n would be solvable, because subgroups of solvable groups are solvable. This contradicts step 9.
  11. Hence SnS_n is not solvable.

Idea

A simple group that is not abelian cannot be solvable: any series must jump from the trivial group straight to the whole group, and that factor is not abelian. For n≥5n \geq 5 the group AnA_n is such a group. A solvable SnS_n would make its subgroup AnA_n solvable.

Reading path

To understand this page from the ground up, read these 121 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 disjoint cycles commute, Proof that the kernel is a normal subgroup, Proof that subgroups of solvable groups are solvable, Proof of strong induction from ordinary induction, Proof of the laws of exponents in a group, Proof that the alternating group is generated by three-cycles, Proof of the universal property of the quotient group, Proof that a homomorphism is injective iff its kernel is trivial, Proof that a subgroup is normal iff its left and right cosets agree, Proof of the first isomorphism theorem, Proof that an intersection of subgroups is a subgroup, Proof of the simplicity of the alternating group by three-cycles, Proof that the sign is a homomorphism, Proof of the decomposition of a permutation into disjoint cycles, Proof of the well-ordering principle from strong induction, Proof of the division algorithm, Proof of the sign of a cycle, Proof of the uniqueness of inverses, Proof that homomorphisms preserve identity and inverses, Proof of the subgroup criterion, Proof of the cancellation laws.

  1. Binary operation definition
  2. Cycle notation concept
  3. Definition by recursion concept
  4. Divisibility definition
  5. Element definition
  6. Even and odd permutations concept
  7. Group concept
  8. Mathematical induction concept
  9. Natural numbers object
  10. Peano axioms concept
  11. Proposition concept
  12. Commutativity definition
  13. Identity element definition
  14. Set definition
  15. Commutator of two elements definition
  16. Conjugate elements definition
  17. Group homomorphism definition
  18. Subgroup definition
  19. Symmetric group definition
  20. Successor function definition
  21. Conjunction definition
  22. Disjunction definition
  23. Implication definition
  24. Predicate definition
  25. Inverse element definition
  26. Empty set object
  27. Family of sets definition
  28. Ordered pair definition
  29. Abelian group definition
  30. Transposition definition
  31. Conjugate subgroup definition
  32. Group isomorphism definition
  33. Intersection definition
  34. Kernel of a homomorphism definition
  35. Union definition
  36. Image of a homomorphism definition
  37. Addition of natural numbers definition
  38. Left coset definition
  39. Proof that an intersection of subgroups is a subgroup proof
  40. Right coset definition
  41. Set-builder notation concept
  42. The successor function is injective statement
  43. Zero is not a successor statement
  44. Biconditional definition
  45. Existential quantifier definition
  46. Universal quantifier definition
  47. Cartesian product definition
  48. Proof of the cancellation laws proof
  49. Proof of the uniqueness of inverses proof
  50. Disjoint sets definition
  51. Normal subgroup definition
  52. Proof of the commutativity of addition by induction proof
  53. Subset definition
  54. Order on the natural numbers definition
  55. Union of a family of sets definition
  56. Axiom of extensionality statement
  57. Uniqueness quantifier definition
  58. Uniqueness of inverses statement
  59. Binary relation definition
  60. Cancellation laws statement
  61. Simple group definition
  62. Partition of a set definition
  63. Commutativity of addition statement
  64. Proof that a subgroup is normal iff its left and right cosets agree proof
  65. Proof of strong induction from ordinary induction proof
  66. Reflexive relation definition
  67. Symmetric relation definition
  68. Transitive relation definition
  69. Proof that homomorphisms preserve identity and inverses proof
  70. Proof of the laws of exponents in a group proof
  71. Proof of the subgroup criterion proof
  72. Function definition
  73. Strong induction statement
  74. Normality via left and right cosets statement
  75. Equivalence relation definition
  76. Homomorphisms preserve identity and inverses statement
  77. Laws of exponents in a group statement
  78. Subgroup criterion statement
  79. Injective function definition
  80. Proof of the well-ordering principle from strong induction proof
  81. Quotient group definition
  82. Proof that disjoint cycles commute proof
  83. Proof that the kernel is a normal subgroup proof
  84. Proof that a homomorphism is injective iff its kernel is trivial proof
  85. Subnormal series definition
  86. Well-ordering principle statement
  87. Proof of the universal property of the quotient group proof
  88. Intersection of subgroups is a subgroup statement
  89. Equivalence class definition
  90. Disjoint cycles commute statement
  91. Solvable group definition
  92. The kernel is a normal subgroup statement
  93. A homomorphism is injective iff its kernel is trivial statement
  94. Proof of the division algorithm proof
  95. Proof that equivalence classes form a partition proof
  96. Subgroup generated by a subset definition
  97. Universal property of the quotient group statement
  98. Division algorithm statement
  99. Commutator subgroup definition
  100. Equivalence classes form a partition statement
  101. Proof of the first isomorphism theorem proof
  102. Derived series definition
  103. Proof of the decomposition of a permutation into disjoint cycles proof
  104. First isomorphism theorem statement
  105. Decomposition of a permutation into disjoint cycles statement
  106. Proof that subgroups of solvable groups are solvable proof
  107. Subgroups of solvable groups are solvable statement
  108. Proof of the decomposition of a permutation into transpositions proof
  109. Decomposition of a permutation into transpositions statement
  110. Sign of a permutation definition
  111. Even and odd permutations definition
  112. Proof of the sign of a cycle proof
  113. Proof that the sign is a homomorphism proof
  114. Sign of a cycle statement
  115. Sign is a homomorphism statement
  116. Alternating group definition
  117. Proof that the alternating group is generated by three-cycles proof
  118. The alternating group is generated by three-cycles statement
  119. Proof of the simplicity of the alternating group by three-cycles proof
  120. Simplicity of the alternating group statement
  121. Proof that the symmetric group on five or more letters is not solvable proof

Where this leads

3 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Solvable_group, Non-example
About this page
TagXNDCSE1YAN
URIm2://XNDCSE1YAN
Kindproof
WritingW0 stub
MSC20F16, 20B30
ProvesThe symmetric group on five or more letters is not solvable
UsesSolvable group, Subnormal series, Simple group, Simplicity of the alternating group, Alternating group, Even and odd permutations, Subgroups of solvable groups are solvable, Symmetric group
Changed4 November 2026

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.