Is the order of the product of two commutative finite-order elements necessarily finite? 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.

If $x$ has order $n$, $y$ has order $m$ and $xy=yx$ then $o(xy) \mid nm$. This follows from observing that 
$$(xy)^{nm}=x^{nm}y^{nm}=e^me^n=e.$$
