| Version 2 |
Version 1 |
| \PMlinkescapeword{Cartesian product} |
\PMlinkescapephrase{Cartesian product} |
| \PMlinkescapeword{finite support} |
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| \PMlinkescapeword{index} |
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| \PMlinkescapeword{pointwise} |
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| Let $I$ be an index set and for each $i\in I$ let $G_i$ be a group. |
Let $I$ be an index set and for each $i\in I$ let $G_i$ be a group. |
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| The (\emph{unrestricted}) \emph{direct product} $\prod_{i\in I}G_i$ |
The (\emph{unrestricted}) \emph{direct product} $\prod_{i\in I}G_i$ |
| 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 makes the Cartesian product into a group. |
It is easily verified that this multiplication makes the Cartesian product into a group. |
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This construction is in fact the categorical direct product in the category of groups.
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This construction is in fact the categorial direct product 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 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 those elements with finite support. That is, |
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\[\bigoplus_{i\in I}G_i=\left\{f\in\prod_{i\in I}G_i\big\mid f(i)=1\hbox{ for all but finitely many }i\in I\right\}.\]
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\[\bigoplus_{i\in I}G_i=\left\{f\in\prod_{i\in I}G_i\\mid f(i)=1\hbox{ for all but finitely many }i\in I\right\}.\]
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| The restricted direct product is also called the \emph{direct sum}, although this usage 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}). |
The restricted direct product is also called the \emph{direct sum}, although this usage 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. |
Confusingly, some authors refer to the restricted direct product as simply the direct product. |
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| Note that if $I$ is finite then the direct product and the restricted direct product are the same. |
Note that if $I$ is finite then the direct product and the restricted direct product are the same. |
