The direct sum of abelian groups is a prototypical example of a direct sum. Given two abelian groups (A, ∗) and (B, ·), their direct sum A ⊕ B is the same as their direct product, i.e. its underlying set is the Cartesian product A × B with the group operation ○ given componentwise:
- (a1, b1) ○ (a2, b2) = (a1 ∗ a2, b1 · b2).
This definition generalizes to direct sums of finitely many abelian groups.
For an infinite family of abelian groups Ai for i ∈ I, the direct sum
is a proper subgroup of the direct product. It consists of the elements such that ai is the identity element of Ai for all but finitely many i.
In this case, the direct sum is indeed the coproduct in the category of abelian groups.
Read more about this topic: Direct Sum
Famous quotes containing the words direct, sum and/or groups:
“England and France, Spain and Portugal, Gold Coast and Slave Coast, all front on this private sea; but no bark from them has ventured out of sight of land, though it is without doubt the direct way to India.”
—Henry David Thoreau (1817–1862)
“Society does not consist of individuals but expresses the sum of interrelations, the relations within which these individuals stand.”
—Karl Marx (1818–1883)
“Under weak government, in a wide, thinly populated country, in the struggle against the raw natural environment and with the free play of economic forces, unified social groups become the transmitters of culture.”
—Johan Huizinga (1872–1945)