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 $\mathbf{Grp}$?
Searching lead to the answer no.
Has anybody a counter-example?
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.