Consider all of these groups have elements of orders . While their orders are . Thus, it does not seem as there is a way to improve , what your wrote.
Another problem I am not sure whether I am missing something or not.
A finite group has elements of orders and , where and are distinct primes. What can you conclude about ?
I said that for some . As a consequence of Lagrange's Theorem (the orders of the elements must divide |G|)
does that seem like the observation they wanted me to make?