Nontrivial free product
= Nontrivial free product
A <free product> $A*B$ is nontrivial as a splitting when both factors differ from the <trivial group>. Alternating words $(ab)^r$ with $a,b\ne1$ in different factors show that every such splitting gives an <infinite group>. The distinct normal forms $ab$ and $ba$ also show that the group is not an <abelian group>.