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
Famous quotes containing the words sums, infinite and/or sets:
“If God lived on earth, people would break his windows.”
—Jewish proverb, quoted in Claud Cockburn, Cockburn Sums Up, epigraph (1981)
“Not till we are lost, in other words not till we have lost the world, do we begin to find ourselves, and realize where we are and the infinite extent of our relations.”
—Henry David Thoreau (18171862)
“Music sets up ladders,
it makes us invisible,
it sets us apart,
it lets us escape;
but from the visible
there is no escape.”
—Hilda Doolittle (18861961)