# What is the intuition behind the direct product and the direct sum of groups being identical for finite groups?

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?

• Direct product is usually denoted by $\times$ while $\otimes$ usually stands for tensor product. – Yanko Oct 8 '18 at 20:54
• It's true for Abelian groups only. (Or.. what is the direct sum of noncommutative groups?) – Berci Oct 8 '18 at 20:54
• @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 – Max Oct 9 '18 at 7:31

## 1 Answer

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).