Math 2.0

Group theory and generalizations Theorem

A finite group is nilpotent iff it is the product of its Sylow subgroups

Builds on Nilpotent groupSylow p-subgroupInternal direct productFinite p-groups are nilpotentFrattini argument

Statement

Let GG be a finite group. The following conditions are equivalent:

  1. GG is nilpotent.
  2. For each prime number pp, GG has exactly one Sylow p-subgroup.
  3. GG is the internal direct product of its Sylow subgroups, one for each prime divisor of ∣G∣|G|.

Remarks

The direction from 3 to 1 holds because finite p-groups are nilpotent and a finite direct product of nilpotent groups is nilpotent. The direction from 1 to 2 uses the Frattini argument. In a nilpotent group, a proper subgroup is strictly smaller than its normaliser. For example, ℤ/12ℤ≅ℤ/4ℤ×ℤ/3ℤ\mathbb{Z}/12\mathbb{Z} \cong \mathbb{Z}/4\mathbb{Z} \times \mathbb{Z}/3\mathbb{Z} is nilpotent, but S3S_3 has three Sylow 22-subgroups, so it is not nilpotent.

Reading path

To understand this page from the ground up, read these 84 pages in order. The foundations come first. The path takes the proof Proof of Lagrange's theorem.

  1. Element definition
  2. Group concept
  3. Left coset concept
  4. Proposition concept
  5. Set definition
  6. Centre of a group definition
  7. Conjugate elements definition
  8. Direct product of groups definition
  9. Group homomorphism definition
  10. Laws of exponents in a group statement
  11. Order of a group definition
  12. Subgroup definition
  13. Symmetric group definition
  14. Conjunction definition
  15. Disjunction definition
  16. Implication definition
  17. Predicate definition
  18. Empty set object
  19. Family of sets definition
  20. Ordered pair definition
  21. Intersection definition
  22. Order of an element definition
  23. Union definition
  24. Group action definition
  25. Conjugate subgroup definition
  26. Set-builder notation concept
  27. Centraliser definition
  28. Left coset definition
  29. Product of two subgroups definition
  30. Right coset definition
  31. Subgroup criterion statement
  32. Existential quantifier definition
  33. Universal quantifier definition
  34. Disjoint sets definition
  35. Cartesian product definition
  36. P-group definition
  37. Action by conjugation example
  38. Orbit definition
  39. Stabiliser definition
  40. Subset definition
  41. Union of a family of sets definition
  42. Normal subgroup definition
  43. Normaliser definition
  44. Index of a subgroup definition
  45. Intersection of subgroups is a subgroup statement
  46. Cosets have the same cardinality as the subgroup statement
  47. Binary relation definition
  48. Partition of a set definition
  49. Sylow p-subgroup definition
  50. Fixed points of an action definition
  51. Orbits partition the set statement
  52. Transitive action definition
  53. Internal direct product definition
  54. Normal core definition
  55. Subgroup generated by a subset definition
  56. Order of a group is the index times the order of the subgroup statement
  57. Normality via left and right cosets statement
  58. Reflexive relation definition
  59. Symmetric relation definition
  60. Transitive relation definition
  61. Conjugacy class definition
  62. Action on left cosets example
  63. Lattice of subgroups concept
  64. Quotient group definition
  65. Equivalence relation definition
  66. Nilpotent group definition
  67. Orbit-stabiliser theorem statement
  68. Correspondence theorem statement
  69. Upper central series definition
  70. Equivalence class definition
  71. Size of a conjugacy class is the index of the centraliser statement
  72. Equivalence classes form a partition statement
  73. Class equation statement
  74. A nontrivial finite p-group has nontrivial centre statement
  75. Proof of Lagrange's theorem proof
  76. Finite p-groups are nilpotent statement
  77. Lagrange's theorem statement
  78. Size of an orbit times the size of the stabiliser statement
  79. Fixed points of a p-group action statement
  80. Cauchy's theorem statement
  81. First Sylow theorem statement
  82. Second Sylow theorem statement
  83. Frattini argument statement
  84. A finite group is nilpotent iff it is the product of its Sylow subgroups statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Nilpotent_group, Properties
  2. Web https://en.wikipedia.org/wiki/Nilpotent_group, Examples
About this page
Tag00Y9N80VSS
URIm2://00Y9N80VSS
Kindstatement, theorem
WritingW0 stub
MSC20F18, 20D15
StatusThis copy of the site has no build data. Run m2 status 00Y9N80VSS.
UsesNilpotent group, Sylow p-subgroup, Internal direct product, Finite p-groups are nilpotent, Frattini argument

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