$\boxed{A_5\text{ is not a nontrivial free product}.}$ If $g\ne1$ and $h\ne1$ 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 $60$.

Solution

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

 Solution (source code)