Math 2.0

Group theory and generalizations Theorem

First Sylow theorem

Builds on Sylow p-subgroupCauchy's theoremP-group

Statement

Let GG be a finite group, let pp be a prime number, and let k≥0k \geq 0 with pk∣∣G∣p^k \mid |G|. Then GG has a subgroup of order pkp^k. In particular, GG has a Sylow p-subgroup.

Remarks

For k=1k = 1, this is Cauchy's theorem. Let pnp^n be the largest power of pp that divides ∣G∣|G|. A proof by Wielandt lets GG act by left multiplication on the subsets of GG of size pnp^n. For example, a group of order 1212 has a subgroup of order 44 and a subgroup of order 33.

History

Ludwig Sylow proved the Sylow theorems in 1872, and he published them in Mathematische Annalen.

Reading path

To understand this page from the ground up, read these 62 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. Subset definition
  35. Stabiliser definition
  36. Union of a family of sets definition
  37. Orbit definition
  38. Index of a subgroup definition
  39. Intersection of subgroups is a subgroup statement
  40. Cosets have the same cardinality as the subgroup statement
  41. Binary relation definition
  42. Sylow p-subgroup definition
  43. Partition of a set definition
  44. Normal core definition
  45. Fixed points of an action definition
  46. Orbits partition the set statement
  47. Transitive action definition
  48. Order of a group is the index times the order of the subgroup statement
  49. Reflexive relation definition
  50. Symmetric relation definition
  51. Transitive relation definition
  52. Action on left cosets example
  53. Equivalence relation definition
  54. Orbit-stabiliser theorem statement
  55. Equivalence class definition
  56. Equivalence classes form a partition statement
  57. Proof of Lagrange's theorem proof
  58. Lagrange's theorem statement
  59. Size of an orbit times the size of the stabiliser statement
  60. Fixed points of a p-group action statement
  61. Cauchy's theorem statement
  62. First Sylow theorem statement

Where this leads

10 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Sylow_theorems, Statement
About this page
Tag0V98WY94B8
URIm2://0V98WY94B8
Kindstatement, theorem
WritingW0 stub
MSC20D20
StatusThis copy of the site has no build data. Run m2 status 0V98WY94B8.
UsesSylow p-subgroup, Cauchy's theorem, P-group

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