Example of a group where $o(a)$ and $o(b)$ are finite but $o(ab)$ is infinite Let G be a group and $a,b \in G$. It is given that $o(a)$ and $o(b)$ are finite. Can you give an example of a group where $o(ab)$ is infinite?

Consider the matrices $$A=\left(\begin{array}[cc] .1 & -1\\ 0 & -1\end{array}\right)\quad\text{and}\quad B=\left(\begin{array}[cc].1&0\\ 0 & -1\end{array}\right).$$ You can check that each have order $2$, but their product gives $$(AB)^n=\left(\begin{array}[cc] .1 & n\\ 0 & 1\end{array}\right),$$a matrix without finite order.
