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:
“The most passionate, consistent, extreme and implacable enemy of the Enlightenment and ... all forms of rationalism ... was Johann Georg Hamann. His influence, direct and indirect, upon the romantic revolt against universalism and scientific method ... was considerable and perhaps crucial.”
—Isaiah Berlin (b. 1909)
“And what is the potential man, after all? Is he not the sum of all that is human? Divine, in other words?”
—Henry Miller (18911980)
“Writers and politicians are natural rivals. Both groups try to make the world in their own images; they fight for the same territory.”
—Salman Rushdie (b. 1947)