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:
“James’s great gift, of course, was his ability to tell a plot in shimmering detail with such delicacy of treatment and such fine aloofness—that is, reluctance to engage in any direct grappling with what, in the play or story, had actually “taken place”Mthat his listeners often did not, in the end, know what had, to put it in another way, “gone on.””
—James Thurber (1894–1961)
“but Overall is beyond me: is the sum of these events
I cannot draw, the ledger I cannot keep, the accounting
beyond the account:”
—Archie Randolph Ammons (b. 1926)
“Instead of seeing society as a collection of clearly defined “interest groups,” society must be reconceptualized as a complex network of groups of interacting individuals whose membership and communication patterns are seldom confined to one such group alone.”
—Diana Crane (b. 1933)