Does o(ab)=lcm(o(a),o(b)) for Group G Elements a & b?

[Artoria]

Suppose G is a group, a,b are two elements of G, does o(ab)=lcm(o(a),o(b))?
o(ab) denotes the order of ab, lcm(o(a),o(b)) denotes the Least Common Multiple of o(a) & o(b).

 

[CompuChip]

It is true when a and b commute, and can easily be proven in that case.
When a and b do not commute, the statement is false. For a counterexample, you could try to find some elements in S₃ (e.g. there are elements a and b such that o(a) = 2, o(b) = 3, o(a, b) = 2).