Difference between revisions of "Direct product"
From Online Dictionary of Crystallography
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| − | In group theory, '''direct product''' of two groups (''G'', *) and (''H'', o), denoted by ''G'' × ''H'' is the | + | In group theory, the '''direct product''' of two groups (''G'', *) and (''H'', o), denoted by ''G'' × ''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'' ∈ ''G'', ''h'' ∈ ''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>. | 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'' | + | 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'' | + | 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'' × ''H''. A relaxation of these conditions gives the [[semidirect product]]. |
[[Category:Fundamental crystallography]] | [[Category:Fundamental crystallography]] | ||
Revision as of 13:14, 13 May 2017
Produit direct (Fr). Direktes Produkt (Ge). Producto directo (Sp). Прямое произведение групп (Ru). Prodotto diretto (It). 直積 (Ja).
In group theory, the direct product of two groups (G, *) and (H, o), denoted by G × 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 ∈ G, h ∈ H};
For abelian groups 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 × H. A relaxation of these conditions gives the semidirect product.