Editing Direct sum of groups (section)

==Generalization to sums over infinite sets==

To describe the above properties in the case where ''G'' is the direct sum of an infinite (perhaps uncountable) set of subgroups, more care is needed.

If ''g'' is an element of the [[cartesian product]] Π{''H''<sub>''i''</sub>} of a set of groups, let ''g''<sub>''i''</sub> be the ''i''th element of ''g'' in the product. The '''external direct sum''' of a set of groups {''H''<sub>''i''</sub>} (written as Σ<sub>'''''E'''''</sub>{''H''<sub>''i''</sub>}) is the subset of Π{''H''<sub>''i''</sub>}, where, for each element ''g'' of Σ<sub>'''''E'''''</sub>{''H''<sub>''i''</sub>}, ''g''<sub>''i''</sub> is the identity <math>e_{H_i}</math> for all but a finite number of ''g''<sub>''i''</sub> (equivalently, only a finite number of ''g''<sub>''i''</sub> are not the identity). The group operation in the external direct sum is pointwise multiplication, as in the usual direct product.

This subset does indeed form a group, and for a finite set of groups {''H''<sub>''i''</sub>} the external direct sum is equal to the direct product.

If ''G'' = Σ''H''<sub>''i''</sub>, then ''G'' is isomorphic to Σ<sub>'''''E'''''</sub>{''H''<sub>''i''</sub>}. Thus, in a sense, the direct sum is an "internal" external direct sum. For each element ''g'' in ''G'', there is a unique finite set ''S'' and a unique set {''h''<sub>''i''</sub> ∈ ''H''<sub>''i''</sub> : ''i'' ∈ ''S''} such that ''g'' = Π {''h''<sub>''i''</sub> : ''i'' in ''S''}.