NOETHER'S PROBLEM FOR ABELIAN EXTENSIONS OF CYCLIC P-GROUPS OF NILPOTENCY CLASS 2

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 prime and let G be a p-group of exponent p(e). 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 if G is any finite p-group of nilpotency class 2 which is an abelian extension of a cyclic group.