Math 2.0

Group theory and generalizations Finite groups Lemma

Frattini argument

Builds on Second Sylow theoremNormaliserProduct of two subgroups

Statement

Let GG be a finite group, let HH be a normal subgroup of GG, and let PP be a Sylow p-subgroup of HH. Then G=NG(P)HG = N_G(P) H, where NG(P)N_G(P) is the normaliser of PP in GG.

Remarks

For g∈Gg \in G, the subgroup gPg−1gPg^{-1} lies in HH, because HH is normal, and it is a Sylow pp-subgroup of HH. By the second Sylow theorem in HH, there is h∈Hh \in H with gPg−1=hPh−1gPg^{-1} = hPh^{-1}, so h−1g∈NG(P)h^{-1}g \in N_G(P). The argument is used in the proof that a finite nilpotent group is the direct product of its Sylow subgroups.

History

Giovanni Frattini used the argument in a paper from 1885. He took it, as he said himself, from a paper of Alfredo Capelli from 1884.

Reading path

To understand this page from the ground up, read these 66 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. Product of two subgroups definition
  26. Right coset definition
  27. Set-builder notation concept
  28. Subgroup criterion statement
  29. Existential quantifier definition
  30. Universal quantifier definition
  31. Cartesian product definition
  32. Disjoint sets definition
  33. P-group definition
  34. Normal subgroup definition
  35. Normaliser definition
  36. Subset definition
  37. Stabiliser definition
  38. Union of a family of sets definition
  39. Orbit definition
  40. Index of a subgroup definition
  41. Intersection of subgroups is a subgroup statement
  42. Cosets have the same cardinality as the subgroup statement
  43. Binary relation definition
  44. Sylow p-subgroup definition
  45. Partition of a set definition
  46. Normal core definition
  47. Fixed points of an action definition
  48. Orbits partition the set statement
  49. Transitive action definition
  50. Order of a group is the index times the order of the subgroup statement
  51. Reflexive relation definition
  52. Symmetric relation definition
  53. Transitive relation definition
  54. Action on left cosets example
  55. Equivalence relation definition
  56. Orbit-stabiliser theorem statement
  57. Equivalence class definition
  58. Equivalence classes form a partition statement
  59. Proof of Lagrange's theorem proof
  60. Lagrange's theorem statement
  61. Size of an orbit times the size of the stabiliser statement
  62. Fixed points of a p-group action statement
  63. Cauchy's theorem statement
  64. First Sylow theorem statement
  65. Second Sylow theorem statement
  66. Frattini argument statement

Where this leads

5 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Frattini's_argument, Statement
  2. Web https://en.wikipedia.org/wiki/Frattini's_argument, lead section
About this page
Tag0QTZDKJEDV
URIm2://0QTZDKJEDV
Aliasesfrattinis-argument
Kindstatement, lemma
WritingW0 stub
MSC20D20
StatusThis copy of the site has no build data. Run m2 status 0QTZDKJEDV.
UsesSecond Sylow theorem, Normaliser, Product of two subgroups
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