supposedly first meant to show that a^m is an element of H for
any natural number m then use Lagrange's Theorem to deduce that the order of any
element must divide the order of the group.
Well... First of all, the subgroup generated by is .
You can claim that , and set out to prove this as follows:
Since , for a random integer
use Euclidean division to obtain for some integers with . Then .
So, all the elements of
are contained in its subset and subgroup, . Therefore, .