Let z be some complex number such that .
Define a matrix by
which means that
so A is unitary.
Furthermore, is given by
since A is a diagonal matrix.
Now assume that k is the smallest positive integer such that . Then .
Furthermore, it is easy to see that (other elements remain 1 or 0 whatever k is).
Therefore, A is of order 6, so one subgroup of order is given by the cyclic group .
Hint:Define a matrix by