If "g" is an element of the cartesian product ∏{"H""i"} of a set of groups, let "g""i" be the "i"th element of "g" in the product. The external direct sum of a set of groups {"H""i"} (written as ∑E{"H""i"}) is the subset of ∏{"H""i"}, where, for each element "g" of ∑E{"H""i"}, "g""i" is the identity formula_28 for all but a finite number of "g""i" (equivalently, only a finite number of "g""i" are not the identity). The group operation in the external direct sum is pointwise multiplication, as in the usual direct product.