Restricted sum

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

Let $(G_i)_{i\in I}$ be a family of groups, and let $(H_i)_{i\in I}$ be a family of groups such that $H_i$ is a subgroup of $G_i$ , for each index $i$ . The subset of $\prod_{i\in I}G_i$ of the $(x_i)_{i\in I}$ for which $x_i \in H_i$ for all but finitely many indices $i$ is the restricted sum of the $G_i$ with respect to the $H_i$ .

When the family $(G_i)_{i \in I}$ is finite, this is identical with the direct sum and direct product. When all but finitely many of the $H_i$ are trivial, the restricted sum of the $G_i$ with respect to the $H_i$ is again the direct sum of the $G_i$ . When all but finitely many of the $H_i$ are equal to their corresponding $G_i$ , the restricted sum is the direct product. When all but finitely many of the $H_i$ are normal subgroups of their corresponding $G_i$ , 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.

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Restricted sum

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