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'direct product and restricted direct product of groups'
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| Title of object: |
direct product and restricted direct product of groups |
| Canonical Name: |
DirectProductAndRestrictedDirectProductOfGroups |
| Type: |
Definition |
| Created on: |
2004-12-11 06:12:19 |
| Modified on: |
2004-12-11 17:20:30 |
| Classification: |
msc:20A99 |
| Defines: |
direct product, restricted direct product, direct product of groups, restricted direct product of groups, direct sum, direct sum of groups |
Revision comment (for changes between this and next version):
Preamble:
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Content:
\PMlinkescapeword{finite support}
\PMlinkescapeword{index}
\PMlinkescapeword{pointwise}
Let $I$ be an index set and for each $i\in I$ let $G_i$ be a group.
The (\emph{unrestricted}) \emph{direct product} $\prod_{i\in I}G_i$
is the \PMlinkname{Cartesian product}{GeneralizedCartesianProduct}
$\prod_{i\in I}G_i$ with multiplication defined pointwise,
that is, for all $f,g\in\prod_{i\in I}G_i$ and all $i\in I$
we have $(fg)(i)=f(i)g(i)$.
It is easily verified that this multiplication makes the \PMlinkescapetext{Cartesian product} into a group.
This construction is in fact the \PMlinkname{categorical direct product}{CategoricalDirectProduct} in the category of groups.
The \emph{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,
\[\bigoplus_{i\in I}G_i=\biggl\{f\in\prod_{i\in I}G_i\biggm| f(i)=1\hbox{ for all but finitely many }i\in I\biggr\}.\]
The restricted direct product is also called the \emph{direct sum}, although this usage is often reserved for the case where all the $G_i$ are abelian (see \PMlinkname{direct sum of modules}{DirectSum} and \PMlinkname{categorical direct sum}{CategoricalDirectSum}).
Confusingly, some authors refer to the restricted direct product as simply the direct product.
Note that if $I$ is finite then the direct product and the restricted direct product are the same. |
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