TITLE: Example of a group where $o(a)$ and $o(b)$ are finite but $o(ab)$ is infinite
QUESTION [5 upvotes]: 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?

REPLY [4 votes]: Every direct plane rotation (of angle $2\theta$) is the composite of two orthogonal reflections with fixed lines making an angle of $\theta$. If $2\theta/2\pi$ is irrational, then the rotation of angle $2\theta$ has infinite order.