Math 2.0

Group theory and generalizations Proposition

The number of Sylow subgroups is the index of the normaliser

Builds on Second Sylow theoremNormaliserOrbit-stabiliser theorem

Statement

Let GG be a finite group, let pp be a prime number, and let PP be a Sylow p-subgroup of GG. Then np=[G:NG(P)]n_p = [G : N_G(P)], where NG(P)N_G(P) is the normaliser of PP.

Remarks

By the second Sylow theorem, GG acts transitively by conjugation on Syl⁡p(G)\operatorname{Syl}_p(G). The stabiliser of PP is NG(P)N_G(P), so the orbit-stabiliser theorem gives the formula. In particular, PP is normal in GG if and only if np=1n_p = 1.

Reading path

To understand this page from the ground up, read these 65 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. Conjugate elements definition
  7. Group homomorphism definition
  8. Laws of exponents in a group statement
  9. Order of a group definition
  10. Subgroup definition
  11. Symmetric group definition
  12. Conjunction definition
  13. Disjunction definition
  14. Implication definition
  15. Predicate definition
  16. Empty set object
  17. Family of sets definition
  18. Ordered pair definition
  19. Order of an element definition
  20. Intersection definition
  21. Union definition
  22. Conjugate subgroup definition
  23. Group action definition
  24. Left coset definition
  25. Right coset definition
  26. Set-builder notation concept
  27. Subgroup criterion statement
  28. Existential quantifier definition
  29. Universal quantifier definition
  30. Cartesian product definition
  31. Disjoint sets definition
  32. P-group definition
  33. Normal subgroup definition
  34. Normaliser definition
  35. Subset definition
  36. Stabiliser definition
  37. Union of a family of sets definition
  38. Orbit definition
  39. Index of a subgroup definition
  40. Intersection of subgroups is a subgroup statement
  41. Cosets have the same cardinality as the subgroup statement
  42. Binary relation definition
  43. Sylow p-subgroup definition
  44. Partition of a set definition
  45. Normal core definition
  46. Fixed points of an action definition
  47. Orbits partition the set statement
  48. Transitive action definition
  49. Order of a group is the index times the order of the subgroup statement
  50. Reflexive relation definition
  51. Symmetric relation definition
  52. Transitive relation definition
  53. Action on left cosets example
  54. Equivalence relation definition
  55. Orbit-stabiliser theorem statement
  56. Equivalence class definition
  57. Equivalence classes form a partition statement
  58. Proof of Lagrange's theorem proof
  59. Lagrange's theorem statement
  60. Size of an orbit times the size of the stabiliser statement
  61. Fixed points of a p-group action statement
  62. Cauchy's theorem statement
  63. First Sylow theorem statement
  64. Second Sylow theorem statement
  65. The number of Sylow subgroups is the index of the normaliser statement

Where this leads

5 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Sylow_theorems, Statement
About this page
Tag09ZJ6CAK7S
URIm2://09ZJ6CAK7S
Kindstatement, proposition
WritingW0 stub
MSC20D20
StatusThis copy of the site has no build data. Run m2 status 09ZJ6CAK7S.
UsesSecond Sylow theorem, Normaliser, Orbit-stabiliser theorem

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