Math 2.0

Group theory and generalizations Lemma

Fixed points of a p-group action

Builds on P-groupFixed points of an actionSize of an orbit times the size of the stabiliserOrbits partition the set

Statement

Let pp be a prime number, let GG be a finite p-group, and let GG act on a finite set XX. Let XGX^G be the set of the fixed points of the action. Then ∣X∣≡∣XG∣(modp)|X| \equiv |X^G| \pmod p.

Remarks

The orbits partition XX, and the orbits with one point are exactly the fixed points. Each other orbit has a size that divides ∣G∣|G| and is greater than 11, so its size is divisible by pp. A consequence: if pp does not divide ∣X∣|X|, then the action has at least one fixed point.

Reading path

To understand this page from the ground up, read these 59 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. Partition of a set definition
  43. Normal core definition
  44. Fixed points of an action definition
  45. Orbits partition the set statement
  46. Transitive action definition
  47. Order of a group is the index times the order of the subgroup statement
  48. Reflexive relation definition
  49. Symmetric relation definition
  50. Transitive relation definition
  51. Action on left cosets example
  52. Equivalence relation definition
  53. Orbit-stabiliser theorem statement
  54. Equivalence class definition
  55. Equivalence classes form a partition statement
  56. Proof of Lagrange's theorem proof
  57. Lagrange's theorem statement
  58. Size of an orbit times the size of the stabiliser statement
  59. Fixed points of a p-group action statement

Where this leads

15 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/P-group, Non-trivial center
  2. Web https://en.wikipedia.org/wiki/Sylow_theorems, Proof of the Sylow theorems
About this page
Tag098D2E7DEH
URIm2://098D2E7DEH
Kindstatement, lemma
WritingW0 stub
MSC20D15, 20B05
StatusThis copy of the site has no build data. Run m2 status 098D2E7DEH.
UsesP-group, Fixed points of an action, Size of an orbit times the size of the stabiliser, Orbits partition the set

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