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单词 DirectProductsOfGroups
释义

direct products of groups


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cartesian productMathworldPlanetmath

Let (Gi)iI be a family of groups.

The unrestricted direct product(or complete direct product, or Cartesian product)iIGiis the Cartesian product (http://planetmath.org/GeneralizedCartesianProduct)iIGi with multiplicationPlanetmathPlanetmath defined pointwise,that is, for all f,giIGi and all iIwe have (fg)(i)=f(i)g(i).It is easily verified that this multiplicationmakes the Cartesian product into a group.This construction is in fact the categorical direct product (http://planetmath.org/CategoricalDirectProduct) in the category of groups.

The restricted direct productPlanetmathPlanetmath iIGi is the subgroupMathworldPlanetmathPlanetmath of iIGi consisting of all those elements with finite support. That is,

iIGi={fiIGi|f(i)=1 for all but finitely many iI}.

The restricted direct product is also called the direct sumMathworldPlanetmathPlanetmath, although this usage is usually reserved for the case where all the Gi are abelianMathworldPlanetmath (see direct sum of modules (http://planetmath.org/DirectSum) and categorical direct sum (http://planetmath.org/CategoricalDirectSum)).

The unqualified term direct productMathworldPlanetmathPlanetmathPlanetmathPlanetmath can refer either to the unrestricted direct product or to the restricted direct product, depending on the author.Note that if I is finite then the unrestricted direct product and the restricted direct product are in fact the same.The direct product of two groups G and H is usually written G×H,or sometimes GH (or GH) if G and H are both abelian.

Titledirect products of groups
Canonical nameDirectProductsOfGroups
Date of creation2013-03-22 14:52:54
Last modified on2013-03-22 14:52:54
Owneryark (2760)
Last modified byyark (2760)
Numerical id17
Authoryark (2760)
Entry typeDefinition
Classificationmsc 20A99
Related topicSubdirectProductOfGroups
Related topicDirectProduct2
Definesdirect product
Definesunrestricted direct product
Definescomplete direct product
Definesrestricted direct product
Definesdirect sum
Definesdirect product of groups
Definesunrestricted direct product of groups
Definesrestricted direct product of groups
Definesdirect sum of groups
DefinesCartesian product of g
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更新时间:2025/5/4 16:09:31