  Let K be any field and G be a finite group. Noether's problem asks whether
the fixed field is rational (=purely transcendental) over K. We will prove that
if G is a non-abelian p-group of order p^n containing a cyclic subgroup of
index p and K is any field containing a primitive p^{n-2}-th root of unity,
then K(G) is rational over K.
