Rationality of finite groups of monomial automorphisms of k(x,y) (Q1092116)

Let \(K=k(x_ 1,x_ 2)\) be a purely transcendental extension field of degree 2 of a field k. Let G be a finite subgroup of the group of automorphisms of K over k. G is said to act monomially if for each \(g\in G\) there exists an \(M\in GL(2, {\mathbb{Z}})\) such that \(g(x_ i)/\prod^{2}_{j=1}x_ j^{M_{i,j}}\in k^{\times}\) for \(i=1,2\). The author proves then that the fixed subfield \(K^ G\) of K is purely transcendental over k.