If g^n is in H, then g^kn is in H for all positive integers k. Let z = the order of g in G. kn = (z, n) mod z for some value of k. Therefore, g^(z,n) is in H. Hence, n = (z, n); n divides z.
Let H be a subgroup of a finite group G. Suppose that g belongs to G and n is the smallest possible integer such that g^n in H. Prove that n divides the order of the element g.
I am sorry I am not sure where to start. Thanks for the help.