Let's try some examples:

if and , then clearly .

if and , then clearly .

You're on the right track, it's just that ord(a) | lcm(ord(a),ord(b)), not the other way around.

Also, I think it should be ord(ab) = lcm(ord(a),ord(b)). Do you have an example where ord(ab) < lcm(ord(a),ord(b))?