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 ∏{Hi} of a set of groups, let gi be the ith element of g in the product. The external direct sum of a set of groups {Hi} (written as ∑E{Hi}) is the subset of ∏{Hi}, where, for each element g of ∑E{Hi}, gi is the identity for all but a finite number of gi (equivalently, only a finite number of gi 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 Hi, the external direct sum is identical to the direct product.
If G = ∑Hi, then G is isomorphic to ∑E{Hi}. 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 unique {hi in Hi : i in S} such that g = ∏ {hi : i in S}.
Read more about this topic: Direct Sum Of Groups
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