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?

The standard example is the infinite dihedral group.
Consider the group of maps on $\mathbf{Z}$ 
$$
D_{\infty} = \{ x \mapsto \pm x + b : b \in \mathbf{Z}  \}.
$$
Consider the maps
$$
\sigma: x \mapsto -x,
\qquad
\tau: x \mapsto -x + 1,
$$
both of order $2$. Their composition
$$
\tau \circ \sigma (x) = \tau(\sigma(x)) : x \mapsto x + 1
$$
has infinite order.
