Conjugacy class - Revision history

BrianMcMahon: Tidied translations.

2017-11-29T09:58:53Z

Tidied translations.

BrianMcMahon: Tidied translations and corrected German (U. Mueller)

2017-11-09T17:16:54Z

Tidied translations and corrected German (U. Mueller)

BrianMcMahon at 10:43, 13 May 2017

2017-05-13T10:43:18Z

BrianMcMahon: Style edits to align with printed edition

2017-05-13T10:42:36Z

Style edits to align with printed edition

MassimoNespolo: Lang

2016-11-26T22:38:44Z

Lang

MassimoNespolo: more logical sequece in the text

2013-05-15T15:54:14Z

more logical sequece in the text

BerndSouvignier at 11:19, 2 April 2009

2009-04-02T11:19:28Z

MassimoNespolo at 06:35, 22 April 2007

2007-04-22T06:35:04Z

MassimoNespolo at 11:47, 27 February 2007

2007-02-27T11:47:34Z

MassimoNespolo at 11:42, 25 February 2007

2007-02-25T11:42:46Z

New page

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 group]]s the concept is trivial, since each class is a set of one element.

[[Category:Fundamental crystallography]]

← Older revision		Revision as of 09:58, 29 November 2017
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−	<font color="blue">Classe de conjugaison</font> (''Fr''). <font color="red">Konjugiertenklasse</font> (''Ge''). <font color="black">Classe coniugata</font> (''It''). <font color="~~brown~~">~~Класс сопряжённости~~</font> (''Ru''). <font color="~~purple~~">~~共役類~~</font> (''Ja''). <font color="green">Clase de conjugación</font> (''Sp'').	+	<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'').

← Older revision		Revision as of 17:16, 9 November 2017
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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'').	+	<font color="blue">Classe de conjugaison</font> (''Fr''). <font color="red">Konjugiertenklasse</font> (''Ge''). <font color="black">Classe coniugata</font> (''It''). <font color="brown">Класс сопряжённости</font> (''Ru''). <font color="purple">共役類</font> (''Ja''). <font color="green">Clase de conjugación</font> (''Sp'').

@@ Line 10: / Line 10: @@
 The equivalence class that contains the element ''g''<sub>1</sub> in ''G'' is
-:::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''}
+:::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.

@@ Line 1: / Line 1: @@
-<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'').
+<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'').
-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:
+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:
-::: g<sub>3</sub>g<sub>1</sub>g<sub>3</sub><sup>-1</sup> = g<sub>2</sub>.
+::: ''g''<sub>3</sub>''g''<sub>1</sub>''g''<sub>3</sub><sup>&minus;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
+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 g<sub>1</sub> in G is
+:::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''}
-:::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}
+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 element forms a class on its own.
 [[Category:Fundamental crystallography]]

← Older revision		Revision as of 22:38, 26 November 2016
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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'')	+	<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'').

@@ Line 6: / Line 6: @@
 ::: 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.
+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 g<sub>1</sub> in G is
@@ Line 14: / Line 15: @@
 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 element forms a class on its own.
+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.
 [[Category:Fundamental crystallography]]

← Older revision		Revision as of 06:35, 22 April 2007
Line 1:		Line 1:
−	<font color="blue">Classe de conjugaison</font> (''Fr''); <font color="red">Konjugationsklasse</font> (''Ge''); <font color="black">Classe coniugata</font> (''It'');	+	<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'')

← Older revision		Revision as of 11:47, 27 February 2007
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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'');
		+
		+
	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:		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: