Restricted external direct product

Definition

Suppose $I$ is an indexing set, and $G_{i},i\in I$ is a family of groups. The restricted direct product or restricted external direct product of the $G_{i}$s, also known as the external direct sum, is defined as follows: it is the subgroup of the external direct product of the $G_{i}$s, comprising those elements for which all but finitely many coordinates are equal to the identity element.

The restricted direct product is denoted by:

$\bigoplus _{i\in I}G_{i}$

When $I$ is finite, the restricted direct product equals the (unrestricted) external direct product.

Equivalence with internal direct product

Further information: equivalence of internal and external direct product

If $G$ is the restricted direct product of the $G_{i},i\in I$, then we can associate, to each $G_{i}$, a normal subgroup $N_{i}$ comprising those elements where all except the $i^{th}$ coordinate are trivial. Then, $G$ is generated by the $N_{i}$s, and each $N_{i}$ intersects trivially the subgroup generated by all the other $N_{j}$s.