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:

$⨁_{i \in I} G_{i}$

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 $i^{t h}$ 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.