# Restricted sum

In group theory, the **restricted sum** is a somewhat obscure extension of the notion of direct sum.

Let be a family of groups, and let be a family of groups such that is a subgroup of , for each index . The subset of of the for which for all but finitely many indices is the *restricted sum of the with respect to the *.

When the family is finite, this is identical with the direct sum and direct product. When all but finitely many of the are trivial, the restricted sum of the with respect to the is again the direct sum of the . When all but finitely many of the are equal to their corresponding , the restricted sum is the direct product. When all but finitely many of the are normal subgroups of their corresponding , the restricted sum is a normal subgroup of the direct product.

## Source

- N. Bourbaki,
*Algebra, Ch. 1–3.*Springer, 1989. ISBN 3-540-64243-9.