In some group G, can we exhibit an example of two elements $x,y$ that
- commute with each other
- have finite order
but whose product $xy$ (or $yx$, since they commute) have infinite order?
I can give an example of when the elements don't commute, say of the permutation group on countable digits, and by defining $f(x) = 1 -x$, $g(x) = 2-x$. Then $f \circ g$ and $g \circ f$ are both infinite order.