Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 104 / 1 / ii / Solution

$\boxed{{\mathbb{Z}}^4\text{ is not a nontrivial free product}.}$ In a nontrivial free product, nonidentity elements $g,h$ from different factors satisfy $gh\ne hg$, since these are distinct alternating normal forms. Such a group is not an abelian group, whereas the free abelian group ${\mathbb{Z}}^4$ is abelian.