Difference between revisions of "Conjugacy class"

Latest revision as of 09:58, 29 November 2017

Classe de conjugaison (Fr). Konjugiertenklasse (Ge). Classe coniugata (It). 共役類 (Ja). Класс сопряжённости (Ru). Clase de conjugación (Sp).

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.

@@ Line 1: / Line 1: @@
-If g<sub>1</sub> and g<sub>2</sub> are two elements of a group G, they are called '''conjugate''' if there exists an element g<sub>3</sub>  in G such that:
+<font color="blue">Classe de conjugaison</font> (''Fr''). <font color="red">Konjugiertenklasse</font> (''Ge''). <font color="black">Classe coniugata</font> (''It''). <font color="purple">共役類</font> (''Ja''). <font color="brown">Класс сопряжённости</font> (''Ru''). <font color="green">Clase de conjugación</font> (''Sp'').
-::: g<sub>3</sub>g<sub>1</sub>g<sub>3</sub><sup>-1</sup> = g<sub>2</sub>.
-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(g<sub>1</sub>) and Cl(g<sub>2</sub>) are equal if and only if g<sub>1</sub> and g<sub>2</sub> are conjugate, and disjoint otherwise.
+If ''g''<sub>1</sub> and ''g''<sub>2</sub> are two elements of a group ''G'', they are called '''conjugate''' if there exists an element ''g''<sub>3</sub>  in ''G'' such that:
-The equivalence class that contains the element g<sub>1</sub> in G is
+::: ''g''<sub>3</sub>''g''<sub>1</sub>''g''<sub>3</sub><sup>&minus;1</sup> = ''g''<sub>2</sub>.
-:::Cl(g<sub>1</sub>) = { g<sub>3</sub>g<sub>1</sub>g<sub>3</sub><sup>-1</sup>| g<sub>3</sub> &isin; G}
+Conjugacy is an equivalence relation and therefore partitions ''G'' into equivalence classes: every element of the group belongs to precisely one conjugacy class.
-and is called the '''conjugacy class''' of g<sub>1</sub>. The '''class number''' of G is the number of conjugacy classes.
+The equivalence class that contains the element ''g''<sub>1</sub> in ''G'' is
-For [[abelian group]]s the concept is trivial, since each class is a set of one element.
+:::Cl(''g''<sub>1</sub>) = { ''g''<sub>3</sub>''g''<sub>1</sub>''g''<sub>3</sub><sup>&minus;1</sup>| ''g''<sub>3</sub> &isin; ''G'' }
+and is called the '''conjugacy class''' of ''g''<sub>1</sub>. The '''class number''' of ''G'' is the number of conjugacy classes.
+The classes Cl(''g''<sub>1</sub>) and Cl(''g''<sub>2</sub>) are equal if and only if ''g''<sub>1</sub> and ''g''<sub>2</sub> are conjugate, and disjoint otherwise.
+For [[Abelian group]]s the concept is trivial, since each element forms a class on its own.
 [[Category:Fundamental crystallography]]

Difference between revisions of "Conjugacy class"

From Online Dictionary of Crystallography

Latest revision as of 09:58, 29 November 2017