NOETHER'S PROBLEM FOR ABELIAN EXTENSIONS OF CYCLIC p-GROUPS

Let K be a field and G 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 (i.e., purely transcendental) over K. The first main result of this article is that K(G) is rational over K for a certain class of p-groups having an abelian subgroup of index p. The second main result is that K(G) is rational over K for any group of order p(5) or p(6) (where p is an odd prime) having an abelian normal subgroup such that its quotient group is cyclic. (In both theorems we assume that if char K not equal p then K contains a primitive p(e)-th root of unity, where p(e) is the exponent of G.)