Rationality of three-dimensional quotients by monomial actions

Let G be a finite 2-group and K be a field satisfying that (i) , and (ii) for any a∈K. If G acts on the rational function field K(x,y,z) by monomial K-automorphisms, then the fixed field KG(x,y,z) is rational (=purely transcendental) over K. Applications of this theorem will be given.