Restricted external direct product
Suppose is an indexing set, and is a family of groups. The restricted direct product or restricted external direct product of the s, also known as the external direct sum, is defined as follows: it is the subgroup of the external direct product of the s, comprising those elements for which all but finitely many coordinates are equal to the identity element.
The restricted direct product is denoted by:
When 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 is the restricted direct product of the , then we can associate, to each , a normal subgroup comprising those elements where all except the coordinate are trivial. Then, is generated by the s, and each intersects trivially the subgroup generated by all the other s.