direct products of groups
Let be a family of groups.
The unrestricted direct product (or complete direct product, or Cartesian product) is the Cartesian product (http://planetmath.org/GeneralizedCartesianProduct) with multiplication defined pointwise, that is, for all and all we have . It is easily verified that this multiplication makes 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 product is also called the direct sum, although this usage is usually reserved for the case where all the are abelian (see direct sum of modules (http://planetmath.org/DirectSum) and categorical direct sum (http://planetmath.org/CategoricalDirectSum)).
The unqualified term direct product can refer either to the unrestricted direct product or to the restricted direct product, depending on the author. Note that if is finite then the unrestricted direct product and the restricted direct product are in fact the same. The direct product of two groups and is usually written , or sometimes (or ) if and are both abelian.
|Title||direct products of groups|
|Date of creation||2013-03-22 14:52:54|
|Last modified on||2013-03-22 14:52:54|
|Last modified by||yark (2760)|
|Defines||unrestricted direct product|
|Defines||complete direct product|
|Defines||restricted direct product|
|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|