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_is, also known as the external direct sum, is defined as follows: it is the subgroup of the external direct product of the G_is, 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_is, and each N_i intersects trivially the subgroup generated by all the other N_js.