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.