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Difference between revisions of "Conjugacy class"

From Online Dictionary of Crystallography

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and is called the '''conjugacy class''' of g<sub>1</sub>. The '''class number''' of G is the number of conjugacy classes.
 
and is called the '''conjugacy class''' of g<sub>1</sub>. The '''class number''' of G is the number of conjugacy classes.
  
For [[abelian group]]s the concept is trivial, since each class is a set of one element.
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For [[abelian group]]s the concept is trivial, since each element forms a class on its own.
  
 
[[Category:Fundamental crystallography]]
 
[[Category:Fundamental crystallography]]

Revision as of 11:19, 2 April 2009

Classe de conjugaison (Fr). Konjugationsklasse (Ge). Classe coniugata (It). 共役類 (Ja)


If g1 and g2 are two elements of a group G, they are called conjugate if there exists an element g3 in G such that:

g3g1g3-1 = g2.

Conjugacy is an equivalence relation and therefore partitions G into equivalence classes: every element of the group belongs to precisely one conjugacy class; the classes Cl(g1) and Cl(g2) are equal if and only if g1 and g2 are conjugate, and disjoint otherwise.

The equivalence class that contains the element g1 in G is

Cl(g1) = { g3g1g3-1| g3 ∈ G}

and is called the conjugacy class of g1. The class number of G is the number of conjugacy classes.

For abelian groups the concept is trivial, since each element forms a class on its own.