Direct product - Revision history

BrianMcMahon: Tidied translations.

2017-11-10T13:45:21Z

Tidied translations.

BrianMcMahon: Style edits to align with printed edition

2017-05-13T13:14:39Z

Style edits to align with printed edition

MassimoNespolo at 09:30, 29 May 2007

2007-05-29T09:30:18Z

New page

Produit direct (''Fr''). Direktes Produkt (''Ge''). Producto directo (''Sp''). Прямое произведение групп (''Ru''). Prodotto diretto (''It''). 直積 (''Ja'').

In group theory, '''direct product''' of two groups (''G'', *) and (''H'', o), denoted by ''G'' × ''H'' is the as set of the elements obtained by taking the [[cartesian product]] of the sets of elements of ''G'' and ''H'': {(''g'', ''h''): ''g'' in ''G'', ''h'' in ''H''};

For [[abelian group]]s which are written additively, it may also be called the ''direct sum'' of two groups, denoted by <math>G \oplus H</math>.

The group obtained in this way has a [[normal subgroup]] isomorphic to ''G'' (given by the elements of the form (''g'', 1)), and one isomorphic to ''H'' (comprising the elements (1, ''h'')).

The reverse also holds: if a group ''K'' contains two normal subgroups ''G'' and ''H'', such that ''K''= ''GH'' and the intersection of ''G'' and ''H'' contains only the identity, then ''K'' = ''G'' x ''H''. A relaxation of these conditions gives the [[semidirect product]].

[[Category:Fundamental crystallography]]

← Older revision		Revision as of 13:45, 10 November 2017
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−	<font color="blue">Produit direct</font> (''Fr''). <font color="red"> Direktes Produkt</font> (''Ge''). <font color="~~green~~">~~Producto directo~~</font> (''Sp''). <font color="~~brown~~">~~Прямое произведение групп~~</font> (''Ru''). <font color="~~black~~">~~Prodotto diretto~~</font> (''It''). <font color="~~purple~~">直積</font> (''Ja'').	+	<font color="blue">Produit direct</font> (''Fr''). <font color="red">Direktes Produkt</font> (''Ge''). <font color="brown">Прямое произведение групп</font> (''Ru''). <font color="black">Prodotto diretto</font> (''It''). <font color="purple">直積</font> (''Ja''). <font color="green">Producto directo</font> (''Sp'').

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-In group theory, '''direct product''' of two groups (''G'', *) and (''H'', o), denoted by ''G'' &times; ''H'' is the as set of the elements obtained by taking the [[cartesian product]] of the sets of elements of ''G'' and ''H'': {(''g'', ''h''): ''g'' in ''G'', ''h'' in ''H''};
+In group theory, the '''direct product''' of two groups (''G'', *) and (''H'', o), denoted by ''G'' &times; ''H'', is the set of the elements obtained by taking the [[Cartesian product]] of the sets of elements of ''G'' and ''H'': {(''g'', ''h''): ''g'' &isin; ''G'', ''h'' &isin; ''H''};
 For [[abelian group]]s which are written additively, it may also be called the ''direct sum'' of two groups, denoted by <math>G \oplus H</math>.
-The group obtained in this way has a [[normal subgroup]] isomorphic to ''G'' (given by the elements of the form (''g'', 1)), and one isomorphic to ''H'' (comprising the elements (1, ''h'')).
+The group obtained in this way has a [[normal subgroup]] isomorphic to ''G'' [given by the elements of the form (''g'', 1)], and one isomorphic to ''H'' [comprising the elements (1, ''h'')].
-The reverse also holds: if a group ''K'' contains two normal subgroups ''G'' and ''H'', such that ''K''= ''GH'' and the intersection of ''G'' and ''H'' contains only the identity, then ''K'' = ''G'' x ''H''. A relaxation of these conditions gives the [[semidirect product]].
+The reverse also holds: if a group ''K'' contains two normal subgroups ''G'' and ''H'', such that ''K''= ''GH'' and the intersection of ''G'' and ''H'' contains only the identity, then ''K'' = ''G'' &times; ''H''. A relaxation of these conditions gives the [[semidirect product]].
 [[Category:Fundamental crystallography]]