Noether's problem for p-groups with a cyclic subgroup of index p2

Let K be any field and G be a finite group. Let G act on the rational function field K(xg: g is an element of G) by K-automorphisms defined by g .x(h) = x(gh) for any g, h is an element of G. Noether's problem asks whether the fixed field K(G) = K(xg: g is an element of G)(G) is rational (= purely transcendental) over K. We will prove that if G is a non-abelian p-group of order p(n) (n >= 3) containing a cyclic subgroup of index p(2) and K is any field containing a primitive p(n-2)-th root of unity, then K(G) is rational over K. As a corollary, if G is a non-abelian p-group of order p(3) and K is a field containing a primitive p-th root of unity, then K(G) is rational. (c) 2010 Elsevier Inc. All rights reserved.