| Version 10 |
Version 9 |
| \PMlinkescapeword{finite support} |
\PMlinkescapeword{finite support} |
| \PMlinkescapeword{index} |
\PMlinkescapeword{index} |
| \PMlinkescapeword{pointwise} |
\PMlinkescapeword{pointwise} |
| \PMlinkescapephrase{Cartesian product} |
\PMlinkescapephrase{Cartesian product} |
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| Let $(G_i)_{i\in I}$ be a family of groups. |
Let $(G_i)_{i\in I}$ be a family of groups. |
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The \emph{unrestricted direct product} (or \emph{Cartesian product}) $\prod_{i\in I}G_i$
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The \emph{unrestricted direct product} (or \emph{Cartesian product}) \$\prod_{i\in I}G_i$
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| is the \PMlinkname{Cartesian product}{GeneralizedCartesianProduct} |
is the \PMlinkname{Cartesian product}{GeneralizedCartesianProduct} |
| $\prod_{i\in I}G_i$ with multiplication defined pointwise, |
$\prod_{i\in I}G_i$ with multiplication defined pointwise, |
| that is, for all $f,g\in\prod_{i\in I}G_i$ and all $i\in I$ |
that is, for all $f,g\in\prod_{i\in I}G_i$ and all $i\in I$ |
| we have $(fg)(i)=f(i)g(i)$. |
we have $(fg)(i)=f(i)g(i)$. |
| It is easily verified that this multiplication |
It is easily verified that this multiplication |
| makes the Cartesian product into a group. |
makes the Cartesian product into a group. |
| This construction is in fact the \PMlinkname{categorical direct product}{CategoricalDirectProduct} in the category of groups. |
This construction is in fact the \PMlinkname{categorical direct product}{CategoricalDirectProduct} in the category of groups. |
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| 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, |
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, |
| \[\bigoplus_{i\in I}G_i=\biggl\{f\in\prod_{i\in I}G_i\biggm| f(i)=1\hbox{ for all but finitely many }i\in I\biggr\}.\] |
\[\bigoplus_{i\in I}G_i=\biggl\{f\in\prod_{i\in I}G_i\biggm| f(i)=1\hbox{ for all but finitely many }i\in I\biggr\}.\] |
| The restricted direct product is also called the \emph{direct sum}, although this usage is usually reserved for the case where all the $G_i$ are abelian (see \PMlinkname{direct sum of modules}{DirectSum} and \PMlinkname{categorical direct sum}{CategoricalDirectSum}). |
The restricted direct product is also called the \emph{direct sum}, although this usage is usually reserved for the case where all the $G_i$ are abelian (see \PMlinkname{direct sum of modules}{DirectSum} and \PMlinkname{categorical direct sum}{CategoricalDirectSum}). |
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| The unqualified term \emph{direct product} can refer either to the unrestricted direct product or to the restricted direct product, depending on the author. |
The unqualified term \emph{direct product} 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. |
Note that if $I$ is finite then the unrestricted direct product and the restricted direct product are in fact the same. |
