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Homedirect products of groups

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# direct products of groups

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 $(fg)(i)=f(i)g(i)$.
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,

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

The restricted direct product is also called the *direct sum*, although this usage is usually reserved for the case where all the $G_{i}$ 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 $I$ is finite then the unrestricted direct product and the restricted direct product are in fact the same.
The direct product of two groups $G$ and $H$ is usually written $G\times H$,
or sometimes $G\oplus H$ (or $G\coprod H$) if $G$ and $H$ are both abelian.

## Mathematics Subject Classification

20A99*no label found*

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