Noether's Problem for p-Groups with an Abelian Subgroup of Index p

Let K be a field and G be a finite group. Let G act on the rational function field K(x(g) : 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. Denote by K (G) the fixed field K (x(g) : g is an element of G)(G). Noether's problem then asks whether K(G) is rational over K. Let p be an odd prime and let G be a p-group of exponent p. Assume also that (i) char K = p > 0, or (ii) char K not equal p and K contains a primitive p(e)-th root of unity. In this paper we prove that K(G) is rational over K for the following two types of groups: (1) G is a finite p-group with an abelian normal subgroup H of index p such that H is a direct product of normal subgroups of G of the type C-pb x (C-p)(e) for some b, c with 1 <= b and 0 <= c; (2) G is any group of order p(5) from the isoclinic families with numbers 1, 2, 3, 4, 8 and 9.