direct products of groups


cartesian productMathworldPlanetmath

Let (Gi)iI be a family of groups.

The unrestricted direct product (or complete direct product, or Cartesian product) iIGi is the Cartesian product ( iIGi with multiplicationPlanetmathPlanetmath defined pointwise, that is, for all f,giIGi and all iI we have (fg)(i)=f(i)g(i). It is easily verified that this multiplication makes the Cartesian product into a group. This construction is in fact the categorical direct product ( 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 ( and categorical direct sum (

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.

Title direct products of groups
Canonical name DirectProductsOfGroups
Date of creation 2013-03-22 14:52:54
Last modified on 2013-03-22 14:52:54
Owner yark (2760)
Last modified by yark (2760)
Numerical id 17
Author yark (2760)
Entry type Definition
Classification msc 20A99
Related topic SubdirectProductOfGroups
Related topic DirectProduct2
Defines direct product
Defines unrestricted direct product
Defines complete direct product
Defines restricted direct product
Defines direct sum
Defines direct product of groups
Defines unrestricted direct product of groups
Defines restricted direct product of groups
Defines direct sum of groups
Defines Cartesian product of g