The direct sum ⊕\oplus versus the cartesian product ×\times

In the case of abelian groups, I have been treating these two set operations as more or less indistinguishable. In early mathematics courses, one normally defines An:=A×A××A; however in, for example, the fundamental theorem of finitely generated abelian groups, we normally write that every such group is isomorphic to one of the form
ZnZr1Zrt
where Zn now means ZZZ.

From an intuition perspective, and in the sense of sets, is this more or less the same as Z×Z××Z? (Bear in mind I am normally using these ideas in relation to homology groups.)

Answer

As long as you restrict to finite index sets, the direct sum and the direct product of commutative groups are identical.

For a general index set I, the direct product of commutative groups {Gi} is the full Cartesian product iIGi, whereas the direct sum iIGi
is the subgroup of the direct product consisting of all tuples {gi} with gi=0
except for finitely many iI.

The coincidence of the direct sum and the direct product in additive categories can also be explained in categorical terms, but I’ll let someone else take a crack at that if they so desire.

Attribution
Source : Link , Question Author : Sputnik , Answer Author : Pete L. Clark

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