Order of element of a group

Question

Is there any group such that $o(a)$ and $o(b)$ is finite for some elements $a,b$ in a group, let's say $G$, but order of $ab$, i.e., $o(ab)$ is not finite.

Answer 1

Consider the infinite dihedral group $D_\infty$, which has presentation $\langle R, F \ | \ F^2 = 1, \ RF = FR^{-1} \rangle$.

Since we have $RF = FR^{-1}$, we also have $(RF)^2 = \text{id}$, so $RF$ has finite order. $F$ also has finite order. However, $(RF)(F) = R$, and the rotation subgroup is infinite.

Answer 2

I remember this example: consider the group $G=GL_{2}(\mathbb{Q})$ and take $$a=\begin{bmatrix} 0 & -1 \\1 & 0 \end{bmatrix} \;\;\;\;\; b=\begin{bmatrix} 0 & 1 \\-1 & -1 \end{bmatrix}.$$

You can easily check that $o(a)=4$, $o(b)=3$, but $o(ab)=\infty$.