Example $\mathrm{ord}(x)$ finite, $\mathrm{ord}(y)$ finite and $\mathrm{ord}(xy)$ infinite Another post had quite a similar question Example of a group with elements $a,b$ such that $\mathrm{ord}(a)=\mathrm{ord}(b)=\mathrm{ord}(ab)=2$. So my question is there is any simple example that can be made for an arbitrary group $G$ and elements $x,y \in G$ such that: $\mathrm{ord}(x) < \infty $, $\mathrm{ord}(y) < \infty$ and $\mathrm{ord}(xy)$ infinite?

Hint: consider two finite cyclic $G$, $H$ group generated by $x$ and $y$ and consider the free product $G*H$, $ord$(x*y)$ is infinite.

See the answer here

Free products of cyclic groups