Math 2.0

statement theorem

Class equation

Statement

Let GG be a finite group with centre Z(G)Z(G). Let a1,…,aka_1, \dots, a_k be one element from each conjugacy class of GG that has more than one element. Then ∣G∣=∣Z(G)∣+∑i=1k[G:CG(ai)]|G| = |Z(G)| + \sum_{i=1}^{k} [G : C_G(a_i)].

Remarks

The conjugacy classes partition GG, and the classes with one element are exactly the elements of Z(G)Z(G). Each term [G:CG(ai)][G : C_G(a_i)] is the size of a class, so it divides ∣G∣|G| and is greater than 11. For S3S_3, the equation is 6=1+3+26 = 1 + 3 + 2. The class equation is the key step in the proof that a nontrivial finite pp-group has a nontrivial centre.

Explanations

L1 human

Count the elements of the group class by class. The central elements form classes of size one, and the other classes have sizes that divide the order of the group. This simple count gives strong facts, for example about groups of prime power order.

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What this needs

To read this node, read these 24 nodes in this order. The foundations come first, and the list ends with this node.

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…statementClass equation, statement, no evidence levelClass equationstatement
Each node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.
  • 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
  24. Class equation statement

What depends on this

4 nodes use this node, directly or through other nodes.

Sources

  1. Web https://en.wikipedia.org/wiki/Conjugacy_class, Conjugacy class equation

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