PlanetMath: direct products of groups

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direct products of groups

(Definition)

Let $(G_i)_{i\in I}$ be a family of groups.

The unrestricted direct product (or complete direct product, or Cartesian product) $\prod_{i\in I}G_i$ is the Cartesian product $\prod_{i\in I}G_i$ with multiplication defined pointwise, that is, for all $f,g\in\prod_{i\in I}G_i$ and all $i\in I$ 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 in the category of groups.

The restricted direct product $\bigoplus_{i\in I}G_i$ is the subgroup of $\prod_{i\in I}G_i$ consisting of all those elements with finite support. That is,

$\displaystyle \bigoplus_{i\in I}G_i=\biggl\{f\in\prod_{i\in I}G_i\biggm\vert f(i)=1\hbox{ for all but finitely many }i\in I\biggr\}.$

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 and categorical direct sum).

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 $G\times H$ , or sometimes $G\oplus H$ (or $G\coprod H$ ) if and are both abelian.

"direct products of groups" is owned by yark.

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Also defines:

direct product, unrestricted direct product, complete direct product, restricted direct product, direct sum, direct product of groups, unrestricted direct product of groups, restricted direct product of groups, direct sum of groups, Cartesian product of groups, complete direct product of groups

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Cross-references: finite, abelian, subgroup, category, multiplication, groups
There are 60 references to this entry.

This is version 14 of direct products of groups, born on 2004-12-11, modified 2007-01-26.
Object id is 6560, canonical name is DirectProductAndRestrictedDirectProductOfGroups.
Accessed 10409 times total.

Classification:

AMS MSC:

20A99 (Group theory and generalizations :: Foundations :: Miscellaneous)

Pending Errata and Addenda

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