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.