Restricted external direct product

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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.