Math 2.0

Group theory and generalizations Proposition

Size of a conjugacy class is the index of the centraliser

Builds on Conjugacy classCentraliserOrbit-stabiliser theoremAction by conjugation

Statement

Let GG be a group and let a∈Ga \in G. Then ∣Cl⁡(a)∣=[G:CG(a)]|\operatorname{Cl}(a)| = [G : C_G(a)], where Cl⁡(a)\operatorname{Cl}(a) is the conjugacy class of aa and CG(a)C_G(a) is its centraliser. In particular, if GG is finite, then ∣Cl⁡(a)∣|\operatorname{Cl}(a)| divides ∣G∣|G|.

Remarks

This is the orbit-stabiliser theorem for the action by conjugation. Directly, gag−1=hah−1gag^{-1} = hah^{-1} if and only if h−1g∈CG(a)h^{-1}g \in C_G(a). For example, in S3S_3 the centraliser of (1 2 3)(1\ 2\ 3) is A3A_3, of index 22, and the class of (1 2 3)(1\ 2\ 3) has 22 elements.

Reading path

To understand this page from the ground up, read these 23 pages in order. The foundations come first.

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Group, concept, no evidence levelGroupconceptCentre of a group, definition, no evidence levelCentre of a groupdefinitionConjugate elements, definition, no evidence levelConjugate elementsdefinitionGroup homomorphism, definition, no evidence levelGroup homomorphismdefinitionSubgroup, definition, no evidence levelSubgroupdefinitionSymmetric group, definition, no evidence levelSymmetric groupdefinitionConjugate subgroup, definition, no evidence levelConjugate subgroupdefinitionGroup action, definition, no evidence levelGroup actiondefinitionCentraliser, definition, no evidence levelCentraliserdefinitionLeft coset, definition, no evidence levelLeft cosetdefinitionSubgroup criterion, statement, no evidence levelSubgroup criterionstatementAction by conjugation, example, no evidence levelAction by conjugationexampleOrbit, definition, no evidence levelOrbitdefinitionNormal subgroup, definition, no evidence levelNormal subgroupdefinitionStabiliser, definition, no evidence levelStabiliserdefinitionIntersection of subgroups is a subgroup, statement, no evidence levelIntersection of subgr…statementOrbits partition the set, statement, no evidence levelOrbits partition the …statementTransitive action, definition, no evidence levelTransitive actiondefinitionNormal core, definition, no evidence levelNormal coredefinitionConjugacy class, definition, no evidence levelConjugacy classdefinitionAction on left cosets, example, no evidence levelAction on left cosetsexampleOrbit-stabiliser theorem, statement, no evidence levelOrbit-stabiliser theo…statementSize of a conjugacy class is the index of the centraliser, statement, no evidence levelSize of a conjugacy c…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Group concept
  2. Centre of a group definition
  3. Conjugate elements definition
  4. Group homomorphism definition
  5. Subgroup definition
  6. Symmetric group definition
  7. Conjugate subgroup definition
  8. Group action definition
  9. Centraliser definition
  10. Left coset definition
  11. Subgroup criterion statement
  12. Action by conjugation example
  13. Orbit definition
  14. Normal subgroup definition
  15. Stabiliser definition
  16. Intersection of subgroups is a subgroup statement
  17. Orbits partition the set statement
  18. Transitive action definition
  19. Normal core definition
  20. Conjugacy class definition
  21. Action on left cosets example
  22. Orbit-stabiliser theorem statement
  23. Size of a conjugacy class is the index of the centraliser statement

Where this leads

5 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Conjugacy_class, Conjugation as a group action, centralizers, and the class equation
About this page
Tag0VK5V9P8G3
URIm2://0VK5V9P8G3
Kindstatement, proposition
WritingW0 stub
MSC20E45
StatusThis copy of the site has no build data. Run m2 status 0VK5V9P8G3.
UsesConjugacy class, Centraliser, Orbit-stabiliser theorem, Action by conjugation

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