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

$\boxed{A_5\text{ is not a nontrivial free product}.}$ If $g\ne 1$ and $h\ne 1$ belong to different factors, the powers $(gh{)}^r$ have distinct normal forms of length $2r$. Every nontrivial free product is therefore an infinite group, whereas the alternating group $A_5$ is a finite group of order $60$.