From Wikipedia: "There is no general formula relating the order of a product ab to the orders of a and b. In fact, it is possible that both a and b have finite order while ab has infinite order, or that both a and b have infinite order while ab has finite order. If ab = ba, we can at least say that ord(ab) divides lcm(ord(a), ord(b)). As a consequence, one can prove that in a finite abelian group, if m denotes the maximum of all the orders of the group's elements, then every element's order divides m."
In fact, we can say more: if (ord(a),ord(b))=1, ord(ab)=ord(a)*ord(b) So suppose the group is abelian. What can we say about order(ab)? Is there any characterization?