Direct Sum - Direct Sum of Abelian Groups

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 A_i for i ∈ I, the direct sum

is a proper subgroup of the direct product. It consists of the elements such that a_i is the identity element of A_i for all but finitely many i.

In this case, the direct sum is indeed the coproduct in the category of abelian groups.