Math 2.0

Group theory and generalizations Definition

Hall subgroup

Builds on GroupSubgroupOrder of a groupIndex of a subgroupCoprime integersPrime numberSylow p-subgroupHall's theorem on solvable groupsSchur–Zassenhaus theorem

Definition

Let GG be a finite group. A Hall subgroup of GG is a subgroup HH whose order and index are coprime: gcd⁡(∣H∣,[G:H])=1\gcd(\abs{H}, [G : H]) = 1.

Let π\pi be a set of prime numbers. A π\pi-number is a positive integer whose prime factors all lie in π\pi. A Hall π\pi-subgroup of GG is a subgroup HH such that ∣H∣\abs{H} is a π\pi-number and [G:H][G : H] has no prime factor in π\pi.

Remarks

A Hall subgroup HH is a Hall π\pi-subgroup for π\pi the set of primes dividing ∣H∣\abs{H}. For π={p}\pi = \set{p}, the Hall π\pi-subgroups are the Sylow pp-subgroups.

Example: A4≤A5A_4 \le A_5. Here ∣A4∣=12\abs{A_4} = 12 and [A5:A4]=5[A_5 : A_4] = 5, and gcd⁡(12,5)=1\gcd(12, 5) = 1, so A4A_4 is a Hall {2,3}\set{2, 3}-subgroup of A5A_5.

Non-example: the subgroup {0,2}\set{0, 2} of ℤ/4ℤ\Z/4\Z. Its order is 22 and its index is 22, and gcd⁡(2,2)=2≠1\gcd(2, 2) = 2 \ne 1.

Hall π\pi-subgroups need not exist: A5A_5 has no subgroup of order 1515 or 2020, so it has no Hall {3,5}\set{3, 5}-subgroup and no Hall {2,5}\set{2, 5}-subgroup. In solvable groups they always exist, by Hall's theorem. A normal Hall subgroup always has a complement, by the Schur–Zassenhaus theorem.

Reading path

To understand this page from the ground up, read these 95 pages in order. The foundations come first. The path takes the proof Proof of Lagrange's theorem.

  1. Abelianisation concept
  2. Commutator subgroup concept
  3. Conjugate elements concept
  4. Conjugate subgroup concept
  5. Coprime integers concept
  6. Element definition
  7. Group concept
  8. Group extension concept
  9. Hall subgroup concept
  10. Index of a subgroup concept
  11. Left coset concept
  12. Normal subgroup concept
  13. Normaliser concept
  14. Order of a group concept
  15. P-group concept
  16. Prime number concept
  17. Proposition concept
  18. Semidirect product concept
  19. Simple group concept
  20. Solvable group concept
  21. Subgroup concept
  22. Subgroup generated by a subset concept
  23. Sylow p-subgroup concept
  24. Transfer homomorphism concept
  25. Set definition
  26. Conjugate elements definition
  27. Group homomorphism definition
  28. Laws of exponents in a group statement
  29. Order of a group definition
  30. Subgroup definition
  31. Symmetric group definition
  32. Schur–Zassenhaus theorem statement
  33. Focal subgroup theorem statement
  34. Conjunction definition
  35. Disjunction definition
  36. Implication definition
  37. Predicate definition
  38. Empty set object
  39. Family of sets definition
  40. Ordered pair definition
  41. Conjugate subgroup definition
  42. Order of an element definition
  43. Group action definition
  44. Intersection definition
  45. Left coset definition
  46. Product of two subgroups definition
  47. Right coset definition
  48. Subgroup criterion statement
  49. Union definition
  50. Set-builder notation concept
  51. Existential quantifier definition
  52. Universal quantifier definition
  53. Cartesian product definition
  54. Disjoint sets definition
  55. Normal subgroup definition
  56. Normaliser definition
  57. P-group definition
  58. Orbit definition
  59. Stabiliser definition
  60. Index of a subgroup definition
  61. Intersection of subgroups is a subgroup statement
  62. Subset definition
  63. Cosets have the same cardinality as the subgroup statement
  64. Union of a family of sets definition
  65. Normal core definition
  66. Binary relation definition
  67. Sylow p-subgroup definition
  68. Orbits partition the set statement
  69. Transitive action definition
  70. Fixed points of an action definition
  71. Order of a group is the index times the order of the subgroup statement
  72. Partition of a set definition
  73. Reflexive relation definition
  74. Symmetric relation definition
  75. Transitive relation definition
  76. Action on left cosets example
  77. Equivalence relation definition
  78. Orbit-stabiliser theorem statement
  79. Equivalence class definition
  80. Equivalence classes form a partition statement
  81. Proof of Lagrange's theorem proof
  82. Lagrange's theorem statement
  83. Converse of Lagrange's theorem fails for the alternating group A4 counterexample
  84. Size of an orbit times the size of the stabiliser statement
  85. Fixed points of a p-group action statement
  86. Third Sylow theorem statement
  87. Cauchy's theorem statement
  88. First Sylow theorem statement
  89. Second Sylow theorem statement
  90. The number of Sylow subgroups is the index of the normaliser statement
  91. Frattini argument statement
  92. The number of Sylow subgroups divides the index statement
  93. Sylow theorems statement
  94. Hall's theorem on solvable groups statement
  95. Hall subgroup definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Hall_subgroup
About this page
TagE09VVB7BJQ
URIm2://E09VVB7BJQ
Kinddefinition
WritingW0 stub
MSC20D20
DefinesHall subgroup
UsesGroup, Subgroup, Order of a group, Index of a subgroup, Coprime integers, Prime number, Sylow p-subgroup, Hall's theorem on solvable groups, Schur–Zassenhaus theorem
Changed2 November 2026

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