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Difference between revisions of "Conjugacy class"

From Online Dictionary of Crystallography

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<font color="blue">Classe de conjugaison</font> (''Fr''). <font color="red">Konjugationsklasse</font> (''Ge''). <font color="black">Classe coniugata</font> (''It''). <font color="purple">共役類</font> (''Ja'')
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<font color="blue">Classe de conjugaison</font> (''Fr''); <font color="red">Konjugationsklasse</font> (''Ge''); <font color="green">Clase de conjugación</font> (''Sp''); <font color="black">Classe coniugata</font> (''It''); <font color="brown">Класс сопряжённости</font> (''Ru''); <font color="purple">共役類</font> (''Ja'').
  
  

Revision as of 22:38, 26 November 2016

Classe de conjugaison (Fr); Konjugationsklasse (Ge); Clase de conjugación (Sp); Classe coniugata (It); Класс сопряжённости (Ru); 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 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.

The classes Cl(g1) and Cl(g2) are equal if and only if g1 and g2 are conjugate, and disjoint otherwise.


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