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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asked Oct 8 '18 at 20:52
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$\begingroup$ Direct product is usually denoted by $\times$ while $\otimes$ usually stands for tensor product. $\endgroup$ – Yanko Oct 8 '18 at 20:54
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1$\begingroup$ It's true for Abelian groups only. (Or.. what is the direct sum of noncommutative groups?) $\endgroup$ – Berci Oct 8 '18 at 20:54
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$\begingroup$ @Berci : direct sum is defined for any family of groups, in the same way as for abelian groups : it's the subgroup of the product where all but finitely many coordinates are $1$ (with multiplicative notation). It's simply not a coproduct, but it still exists $\endgroup$ – Maxime Ramzi Oct 9 '18 at 7:31
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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).
