Semidirect product - Revision history

BrianMcMahon: Tidied translations.

2017-12-15T11:32:16Z

Tidied translations.

BrianMcMahon: Tidied translations.

2017-11-20T08:42:21Z

Tidied translations.

BrianMcMahon: Style edits to align with printed edition

2017-05-17T10:32:17Z

Style edits to align with printed edition

MassimoNespolo: broken link

2008-12-21T18:31:25Z

broken link

MassimoNespolo: link

2007-05-29T10:00:00Z

link

MassimoNespolo at 09:40, 29 May 2007

2007-05-29T09:40:28Z

New page

Produit semi-direct (''Fr''). Semidirektes Produkt (''Ge''). Producto semidirecto (''Sp''). Полупрямое произведение (''Ru''). Prodotto semidiretto (''It''). 準直積 (''Ja'').

In group theory, a '''semidirect product''' describes a particular way in which a group can be put together from two subgroups, one of which is [[normal subgroup|normal]].

Let ''G'' be a group, ''N'' a [[normal subgroup]] of ''G'' (i.e., ''N'' ◁ ''G'') and ''H'' a [[subgroup]] of ''G''. ''G'' is a '''semidirect product''' of ''N'' and ''H'' if there exists a [[homomorphism]] ''G'' → ''H'' which is the identity on ''H'' and whose [[kernel (algebra)|kernel]] is ''N''. This is equivalent to say that:
* ''G'' = ''NH'' and ''N'' ∩ ''H'' = {1} (where "1" is identity element of ''G''　)
* ''G'' = ''HN'' and ''N'' ∩ ''H'' = {1}
* Every element of ''G'' can be written as a unique product of an element of ''N'' and an element of ''H''
* Every element of ''G'' can be written as a unique product of an element of ''H'' and an element of ''N''

One also says that "''G'' ''splits'' over ''N''".

[[Category:Fundamental crystallography]]

← Older revision		Revision as of 11:32, 15 December 2017
Line 1:		Line 1:
−	<font color="blue">Produit semi-direct</font> (''Fr''). <font color="red">Semidirektes Produkt</font> (''Ge''). <font color="~~brown~~">~~Полупрямое произведение~~</font> (''Ru''). <font color="~~black~~">~~Prodotto semidiretto~~</font> (''It''). <font color="~~purple~~">~~準直積~~</font> (''Ja''). <font color="green">Producto semidirecto</font> (''Sp'').	+	<font color="blue">Produit semi-direct</font> (''Fr''). <font color="red">Semidirektes Produkt</font> (''Ge''). <font color="black">Prodotto semidiretto</font> (''It''). <font color="purple">準直積</font> (''Ja''). <font color="brown">Полупрямое произведение</font> (''Ru''). <font color="green">Producto semidirecto</font> (''Sp'').

← Older revision		Revision as of 08:42, 20 November 2017
Line 1:		Line 1:
−	<font color="blue">Produit semi-direct</font> (''Fr''). <font color="red">Semidirektes Produkt</font> (''Ge''). <font color="~~green~~">~~Producto semidirecto~~</font> (''Sp''). <font color="~~brown~~">~~Полупрямое произведение~~</font> (''Ru''). <font color="~~black~~">~~Prodotto semidiretto~~</font> (''It''). <font color="~~purple~~">~~準直積~~</font> (''Ja'').	+	<font color="blue">Produit semi-direct</font> (''Fr''). <font color="red">Semidirektes Produkt</font> (''Ge''). <font color="brown">Полупрямое произведение</font> (''Ru''). <font color="black">Prodotto semidiretto</font> (''It''). <font color="purple">準直積</font> (''Ja''). <font color="green">Producto semidirecto</font> (''Sp'').
		+

	In group theory, a '''semidirect product''' describes a particular way in which a group can be put together from two subgroups, one of which is [[normal subgroup\|normal]].		In group theory, a '''semidirect product''' describes a particular way in which a group can be put together from two subgroups, one of which is [[normal subgroup\|normal]].

@@ Line 1: / Line 1: @@
 <font color="blue">Produit semi-direct</font> (''Fr''). <font color="red">Semidirektes Produkt</font> (''Ge''). <font color="green">Producto semidirecto</font> (''Sp''). <font color="brown">Полупрямое произведение</font> (''Ru''). <font color="black">Prodotto semidiretto</font> (''It''). <font color="purple">準直積</font> (''Ja'').
 In group theory, a '''semidirect product''' describes a particular way in which a group can be put together from two subgroups, one of which is [[normal subgroup|normal]].
-Let ''G'' be a group, ''N'' a [[normal subgroup]] of ''G'' (i.e., ''N'' &#x25C1; ''G'') and ''H'' a [[subgroup]] of ''G''. ''G'' is a '''semidirect product''' of ''N'' and ''H'' if there exists a [[group homomorphism|homomorphism]] ''G'' &rarr; ''H'' which is the identity on ''H'' and whose [[Group homomorphism|kernel]] is ''N''. This is equivalent to say that:
+Let ''G'' be a group, ''N'' a [[normal subgroup]] of ''G'' (''i.e.'' ''N'' &#x25C1; ''G'') and ''H'' a [[subgroup]] of ''G''. ''G'' is a '''semidirect product''' of ''N'' and ''H'' if there exists a [[group homomorphism|homomorphism]] ''G'' &rarr; ''H'' which is the identity on ''H'' and whose [[Group homomorphism|kernel]] is ''N''. This is equivalent to saying that:
-* ''G'' = ''NH'' and ''N'' &cap; ''H'' = {1} (where "1" is identity element of ''G''　)
+* ''G'' = ''NH'' and ''N'' &cap; ''H'' = {1} (where '1' is the identity element of ''G'').
-* ''G'' = ''HN'' and ''N'' &cap; ''H'' = {1}
+* ''G'' = ''HN'' and ''N'' &cap; ''H'' = {1}.
-* Every element of ''G'' can be written as a unique product of an element of ''N'' and an element of ''H''
+* Every element of ''G'' can be written as a unique product of an element of ''N'' and an element of ''H''.
-* Every element of ''G'' can be written as a unique product of an element of ''H'' and an element of ''N''
+* Every element of ''G'' can be written as a unique product of an element of ''H'' and an element of ''N''.
-One also says that "''G'' ''splits'' over ''N''".
+One also says that `''G'' ''splits'' over ''N'''.
 [[Category:Fundamental crystallography]]

@@ Line 4: / Line 4: @@
 In group theory, a '''semidirect product''' describes a particular way in which a group can be put together from two subgroups, one of which is [[normal subgroup|normal]].
-Let ''G'' be a group, ''N'' a [[normal subgroup]] of ''G'' (i.e., ''N'' &#x25C1; ''G'') and ''H'' a [[subgroup]] of ''G''. ''G'' is a '''semidirect product''' of ''N'' and ''H'' if there exists a [[group homomorphism|homomorphism]] ''G'' &rarr; ''H'' which is the identity on ''H'' and whose [[kernel (algebra)|kernel]] is ''N''. This is equivalent to say that:
+Let ''G'' be a group, ''N'' a [[normal subgroup]] of ''G'' (i.e., ''N'' &#x25C1; ''G'') and ''H'' a [[subgroup]] of ''G''. ''G'' is a '''semidirect product''' of ''N'' and ''H'' if there exists a [[group homomorphism|homomorphism]] ''G'' &rarr; ''H'' which is the identity on ''H'' and whose [[Group homomorphism|kernel]] is ''N''. This is equivalent to say that:
 * ''G'' = ''NH'' and ''N'' &cap; ''H'' = {1} (where "1" is identity element of ''G''　)
 * ''G'' = ''HN'' and ''N'' &cap; ''H'' = {1}