I have seen the statement that the direct product of groups $G_1\otimes G_2 \otimes … \otimes G_N$ is identical to the direct sum of groups $G_1\oplus G_2 \oplus … \oplus G_N$ for finite $N$.
What is the intuition about this being true?
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It's basically the fact that finite coproducts and products coincide in the category of Abelian groups.
There's a deep connection of these coincidences with addition, as discussed here in details.
Briefly, for any category with zero morphisms:
There (naturally) is a commutative monoid structure ('addition') on the set of morphisms $A\to B$ for any objects $A,B\ $ iff$\ $ finite coproducts and products coincide
Now, pointwise addition of homomorphisms $f,g:A\to B$ produces again a homomorphism $A\to B\ $ (if $B$ is commutative), and thus it gives rise to a commutative monoid structure (with inverses, i.e. an Abelian group).