Math 2.0

Group theory and generalizations Proof

Proof that the size of a conjugacy class is the index of the centraliser

Builds on Action by conjugationConjugacy classCentraliserOrbitStabiliserOrbit-stabiliser theoremSize of an orbit times the size of the stabiliser

Proof

We apply the orbit-stabiliser theorem to the action by conjugation, g⋅x=gxg−1g \cdot x = g x g^{-1}.

  1. The orbit of aa under conjugation is {gag−1:g∈G}\set{g a g^{-1} : g \in G}. By the definition of a conjugacy class, this is Cl⁡(a)\operatorname{Cl}(a).
  2. The stabiliser of aa is {g∈G:gag−1=a}\set{g \in G : g a g^{-1} = a}. Multiplying on the right by gg, or by g−1g^{-1}, shows gag−1=ag a g^{-1} = a if and only if ga=agg a = a g.
  3. So the stabiliser of aa is {g∈G:ga=ag}\set{g \in G : ga = ag}, which is the centraliser CG(a)C_G(a).
  4. By the orbit-stabiliser theorem and steps 1 and 3, ∣Cl⁡(a)∣=[G:CG(a)]\abs{\operatorname{Cl}(a)} = [G : C_G(a)].
  5. If GG is finite, the orbit size formula shows that the size of each orbit divides ∣G∣\abs{G}. By step 1, ∣Cl⁡(a)∣\abs{\operatorname{Cl}(a)} divides ∣G∣\abs{G}.

Idea

Conjugation is an action, its orbits are the conjugacy classes, and the elements that fix aa are those that commute with aa. The orbit-stabiliser theorem then turns the class size into the index of the centraliser.

Reading path

To understand this page from the ground up, read these 82 pages in order. The foundations come first. The path takes the proofs Proof that equivalence classes form a partition, Proof that the size of an orbit times the size of the stabiliser is the order of the group, Proof that an intersection of subgroups is a subgroup, Proof of the orbit-stabiliser theorem, Proof that the order of a finite group is the index times the order of a subgroup, Proof of the uniqueness of inverses, Proof of Lagrange's theorem, Proof that cosets have the same cardinality as the subgroup, Proof of the subgroup criterion, Proof that orbits partition the set.

  1. Binary operation definition
  2. Element definition
  3. Group concept
  4. Left coset concept
  5. Proposition concept
  6. Identity element definition
  7. Set definition
  8. Centre of a group definition
  9. Conjugate elements definition
  10. Group homomorphism definition
  11. Order of a group definition
  12. Subgroup definition
  13. Symmetric group definition
  14. Conjunction definition
  15. Disjunction definition
  16. Implication definition
  17. Predicate definition
  18. Inverse element definition
  19. Empty set object
  20. Family of sets definition
  21. Ordered pair definition
  22. Intersection definition
  23. Union definition
  24. Proof that an intersection of subgroups is a subgroup proof
  25. Group action definition
  26. Conjugate subgroup definition
  27. Centraliser definition
  28. Left coset definition
  29. Right coset definition
  30. Set-builder notation concept
  31. Biconditional definition
  32. Existential quantifier definition
  33. Universal quantifier definition
  34. Proof of the uniqueness of inverses proof
  35. Disjoint sets definition
  36. Cartesian product definition
  37. Action by conjugation example
  38. Stabiliser definition
  39. Subset definition
  40. Normal subgroup definition
  41. Orbit definition
  42. Union of a family of sets definition
  43. Axiom of extensionality statement
  44. Index of a subgroup definition
  45. Uniqueness quantifier definition
  46. Uniqueness of inverses statement
  47. Binary relation definition
  48. Partition of a set definition
  49. Transitive action definition
  50. Reflexive relation definition
  51. Symmetric relation definition
  52. Transitive relation definition
  53. Proof of the subgroup criterion proof
  54. Function definition
  55. Equivalence relation definition
  56. Codomain definition
  57. Injective function definition
  58. Subgroup criterion statement
  59. Surjective function definition
  60. Intersection of subgroups is a subgroup statement
  61. Equivalence class definition
  62. Proof of the orbit-stabiliser theorem proof
  63. Bijective function definition
  64. Proof that equivalence classes form a partition proof
  65. Normal core definition
  66. Equinumerous sets definition
  67. Equivalence classes form a partition statement
  68. Action on left cosets example
  69. Cardinality definition
  70. Proof that orbits partition the set proof
  71. Orbit-stabiliser theorem statement
  72. Proof that cosets have the same cardinality as the subgroup proof
  73. Orbits partition the set statement
  74. Conjugacy class definition
  75. Cosets have the same cardinality as the subgroup statement
  76. Proof of Lagrange's theorem proof
  77. Proof that the order of a finite group is the index times the order of a subgroup proof
  78. Lagrange's theorem statement
  79. Order of a group is the index times the order of the subgroup statement
  80. Proof that the size of an orbit times the size of the stabiliser is the order of the group proof
  81. Size of an orbit times the size of the stabiliser statement
  82. Proof that the size of a conjugacy class is the index of the centraliser proof

Where this leads

32 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Conjugacy_class, Conjugacy class equation
About this page
TagGQ2PGJ9RJC
URIm2://GQ2PGJ9RJC
Kindproof
WritingW0 stub
MSC20E45
ProvesSize of a conjugacy class is the index of the centraliser
UsesAction by conjugation, Conjugacy class, Centraliser, Orbit, Stabiliser, Orbit-stabiliser theorem, Size of an orbit times the size of the stabiliser
Changed4 November 2026

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