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
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.