Abelian group - Revision history

BrianMcMahon at 15:28, 27 November 2017

2017-11-27T15:28:01Z

BrianMcMahon: Tidied translations.

2017-11-08T17:27:00Z

Tidied translations.

MassimoNespolo: languages

2017-10-13T14:44:12Z

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BrianMcMahon: Style edits to align with printed edition

2017-05-12T10:09:33Z

Style edits to align with printed edition

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2007-05-29T09:10:02Z

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MassimoNespolo at 11:42, 27 February 2007

2007-02-27T11:42:40Z

MassimoNespolo at 11:46, 25 February 2007

2007-02-25T11:46:07Z

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An '''abelian group''', also called a ''commutative group'', is a group (G, * ) such that g1 * g2 = g2 * g1 for all g1 and g2 in G, where * is a binary operation in G. This means that the order in which the binary operation is performed does not matter, and any two elements of the group commute.

Groups that are not commutative are called non-abelian (rather than non-commutative).

Abelian groups are named after Niels Henrik Abel.

[[Category:Fundamental crystallography]]

@@ Line 1: / Line 1: @@
 <font color="orange">زمرة أبيلية</font> (''Ar''). <font color="blue">Groupe abélien</font> (''Fr''). <font color="red">Abelsche Gruppe</font> (''Ge''). <font color="black">Gruppo abeliano</font> (''It''). <font color="purple">アーベル群</font> (''Ja''). <font color="brown">Абелева группа</font> (''Ru''). <font color="green">Grupo abeliano</font> (''Sp'').
-An '''abelian group''', also called a ''commutative group'', is a group (''G'', * ) such that ''g''<sub>1</sub> * ''g''<sub>2</sub> = ''g''<sub>2</sub> * ''g''<sub>1</sub> for all ''g''<sub>1</sub> and ''g''<sub>2</sub> in ''G'', where * is a [[binary operation]] in ''G''. This means that the order in which the binary operation is performed does not matter, and any two elements of the group commute.
+An '''Abelian group''', also called a ''commutative group'', is a group (''G'', * ) such that ''g''<sub>1</sub> * ''g''<sub>2</sub> = ''g''<sub>2</sub> * ''g''<sub>1</sub> for all ''g''<sub>1</sub> and ''g''<sub>2</sub> in ''G'', where * is a [[binary operation]] in ''G''. This means that the order in which the binary operation is performed does not matter, and any two elements of the group commute.
 Groups that are not commutative are called non-abelian (rather than non-commutative).

← Older revision		Revision as of 17:27, 8 November 2017
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−	<font color="orange">زمرة أبيلية</font> (''Ar''); <font color="blue">Groupe abélien</font> (''Fr''); <font color="red">Abelsche Gruppe</font> (''Ge''); <font color="black">Gruppo abeliano</font> (''It''); <font color="purple">アーベル群</font> (''Ja''); <font color="brown">Абелева группа</font> (''Ru''); <font color="green">Grupo abeliano</font> (''Sp'').	+	<font color="orange">زمرة أبيلية</font> (''Ar''). <font color="blue">Groupe abélien</font> (''Fr''). <font color="red">Abelsche Gruppe</font> (''Ge''). <font color="black">Gruppo abeliano</font> (''It''). <font color="purple">アーベル群</font> (''Ja''). <font color="brown">Абелева группа</font> (''Ru''). <font color="green">Grupo abeliano</font> (''Sp'').

	An '''abelian group''', also called a ''commutative group'', is a group (''G'', * ) such that ''g''<sub>1</sub> * ''g''<sub>2</sub> = ''g''<sub>2</sub> * ''g''<sub>1</sub> for all ''g''<sub>1</sub> and ''g''<sub>2</sub> in ''G'', where * is a [[binary operation]] in ''G''. This means that the order in which the binary operation is performed does not matter, and any two elements of the group commute.		An '''abelian group''', also called a ''commutative group'', is a group (''G'', * ) such that ''g''<sub>1</sub> * ''g''<sub>2</sub> = ''g''<sub>2</sub> * ''g''<sub>1</sub> for all ''g''<sub>1</sub> and ''g''<sub>2</sub> in ''G'', where * is a [[binary operation]] in ''G''. This means that the order in which the binary operation is performed does not matter, and any two elements of the group commute.

← Older revision		Revision as of 14:44, 13 October 2017
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−	<font color="blue">Groupe abélien</font> (''Fr''). <font color="red">Abelsche Gruppe</font> (''Ge''). <font color="~~green~~">~~Grupo~~ abeliano</font> (''Sp''). <font color="~~black~~">~~Gruppo abeliano~~</font> (''It''). <font color="~~purple~~">~~アーベル群~~</font> (''Ja'').	+	<font color="orange">زمرة أبيلية</font> (''Ar''); <font color="blue">Groupe abélien</font> (''Fr''); <font color="red">Abelsche Gruppe</font> (''Ge''); <font color="black">Gruppo abeliano</font> (''It''); <font color="purple">アーベル群</font> (''Ja''); <font color="brown">Абелева группа</font> (''Ru''); <font color="green">Grupo abeliano</font> (''Sp'').

	An '''abelian group''', also called a ''commutative group'', is a group (''G'', * ) such that ''g''<sub>1</sub> * ''g''<sub>2</sub> = ''g''<sub>2</sub> * ''g''<sub>1</sub> for all ''g''<sub>1</sub> and ''g''<sub>2</sub> in ''G'', where * is a [[binary operation]] in ''G''. This means that the order in which the binary operation is performed does not matter, and any two elements of the group commute.		An '''abelian group''', also called a ''commutative group'', is a group (''G'', * ) such that ''g''<sub>1</sub> * ''g''<sub>2</sub> = ''g''<sub>2</sub> * ''g''<sub>1</sub> for all ''g''<sub>1</sub> and ''g''<sub>2</sub> in ''G'', where * is a [[binary operation]] in ''G''. This means that the order in which the binary operation is performed does not matter, and any two elements of the group commute.

@@ Line 1: / Line 1: @@
-<font color="blue">Groupe abélien</font> (''Fr''); <font color="red">Abelsche Gruppe</font> (''Ge''); <font color="green">Grupo abeliano</font> (''Sp''); <font color="black">Gruppo abeliano</font> (''It''); <font color="purple">アーベル群</font> (''Ja'')
+<font color="blue">Groupe abélien</font> (''Fr''). <font color="red">Abelsche Gruppe</font> (''Ge''). <font color="green">Grupo abeliano</font> (''Sp''). <font color="black">Gruppo abeliano</font> (''It''). <font color="purple">アーベル群</font> (''Ja'').
+An '''abelian group''', also called a ''commutative group'', is a group (''G'', * ) such that ''g''<sub>1</sub> * ''g''<sub>2</sub> = ''g''<sub>2</sub> * ''g''<sub>1</sub> for all ''g''<sub>1</sub> and ''g''<sub>2</sub> in ''G'', where * is a [[binary operation]] in ''G''. This means that the order in which the binary operation is performed does not matter, and any two elements of the group commute.
 Groups that are not commutative are called non-abelian (rather than non-commutative).

← Older revision		Revision as of 09:10, 29 May 2007
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−	An '''abelian group''', also called a ''commutative group'', is a group (G, * ) such that g<sub>1</sub> * g<sub>2</sub> = g<sub>2</sub> * g<sub>1</sub> for all g<sub>1</sub> and g<sub>2</sub> in G, where * is a binary operation in G. This means that the order in which the binary operation is performed does not matter, and any two elements of the group commute.	+	An '''abelian group''', also called a ''commutative group'', is a group (G, * ) such that g<sub>1</sub> * g<sub>2</sub> = g<sub>2</sub> * g<sub>1</sub> for all g<sub>1</sub> and g<sub>2</sub> in G, where * is a [[binary operation]] in G. This means that the order in which the binary operation is performed does not matter, and any two elements of the group commute.

	Groups that are not commutative are called non-abelian (rather than non-commutative).		Groups that are not commutative are called non-abelian (rather than non-commutative).

← Older revision		Revision as of 11:42, 27 February 2007
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		+	<font color="blue">Groupe abélien</font> (''Fr''); <font color="red">Abelsche Gruppe</font> (''Ge''); <font color="green">Grupo abeliano</font> (''Sp''); <font color="black">Gruppo abeliano</font> (''It''); <font color="purple">アーベル群</font> (''Ja'')
		+
		+
	An '''abelian group''', also called a ''commutative group'', is a group (G, * ) such that g<sub>1</sub> * g<sub>2</sub> = g<sub>2</sub> * g<sub>1</sub> for all g<sub>1</sub> and g<sub>2</sub> in G, where * is a binary operation in G. This means that the order in which the binary operation is performed does not matter, and any two elements of the group commute.		An '''abelian group''', also called a ''commutative group'', is a group (G, * ) such that g<sub>1</sub> * g<sub>2</sub> = g<sub>2</sub> * g<sub>1</sub> for all g<sub>1</sub> and g<sub>2</sub> in G, where * is a binary operation in G. This means that the order in which the binary operation is performed does not matter, and any two elements of the group commute.