Math 2.0

Group theory and generalizations Definition

Conjugacy class

Builds on Conjugate elementsOrbits partition the set

Definition

Let GG be a group and let a∈Ga \in G. The conjugacy class of aa is the set Cl⁡(a)={gag−1:g∈G}\operatorname{Cl}(a) = \{gag^{-1} : g \in G\}.

Remarks

The conjugacy classes are the equivalence classes of the relation "is conjugate to", so they partition GG. They are the orbits of the action by conjugation. The class of aa is {a}\{a\} if and only if aa lies in the centre of GG. In the symmetric group SnS_n, the conjugacy classes are given by the cycle types.

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Group, concept, no evidence levelGroupconceptConjugate elements, definition, no evidence levelConjugate elementsdefinitionGroup homomorphism, definition, no evidence levelGroup homomorphismdefinitionSymmetric group, definition, no evidence levelSymmetric groupdefinitionGroup action, definition, no evidence levelGroup actiondefinitionOrbit, definition, no evidence levelOrbitdefinitionOrbits partition the set, statement, no evidence levelOrbits partition the …statementConjugacy class, definition, no evidence levelConjugacy classdefinition
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  1. Group concept
  2. Conjugate elements definition
  3. Group homomorphism definition
  4. Symmetric group definition
  5. Group action definition
  6. Orbit definition
  7. Orbits partition the set statement
  8. Conjugacy class definition

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6 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Conjugacy_class, Definition
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DefinesConjugacy class
UsesConjugate elements, Orbits partition the set

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