# Coproducts in $\mathbf{Grp}$ and in $\mathbf{Ab}$

In the category $\mathbf{Ab}$ of abelian groups the coproduct of two groups is the direct sum $A\oplus B$, which is the same as the cartesian product $A\times B$.

I wonder if this is also true in the category $\mathbf{Grp}$ of groups. So: Let $A,B$ be two abelian groups. Is $A\times B$ in general a coproduct product in $\mathbf{Grp}$?

In the category of groups, the coproduct is the free product of groups. For example, $A = B = \mathbb Z$ has $F_2$ (free group on $2$ generators) as coproduct and $\mathbb Z^2$ as product.
• It might be worth noting that coproduct in $\bf{Ab}$ is $\Bbb Z\oplus \Bbb Z$ which is free abelian group on $2$ generators. This is not coincidence, since $\Bbb Z$ is free group over $1$ generator, and free objects commute with coproducts. Thus, coproduct of any number of copies of $\Bbb Z$ must be free object on that same number of generators. That is why we get different coproducts in $\bf{Grp}$ and $\bf{Ab}$, since construction of free objects is not same. – Ennar Nov 13 '16 at 14:38
• Without knowing the fact that the coproduct is the free product: Can one somehow prove that $\mathbb{Z}\times\mathbb{Z}$ is not a coproduct in the category of groups? – user369147 Nov 13 '16 at 14:40
• @user114179, yes. Just write down appropriate diagram for coproduct and show that $\Bbb Z\times\Bbb Z$ does not satisfy the universal property by finding counterexample. – Ennar Nov 13 '16 at 14:43
• if you take just about any group $\notin Ab$, you can show the coproduct in $Ab$ does not satisfy the universal property in $Grp$. – Matematleta Nov 13 '16 at 15:01