Noether's problem for some subgroups of S₁₄: The modular case

Let G be a subgroup of S_n, the symmetric group of degree n. For any field k, G acts naturally on the rational function field k(x₁,⋯,x_n) via k-automorphisms defined by σ⋅x_i:=x_σ⋅i for any σ∈G and 1≤i≤n. In this article, we will show that if G is a solvable transitive subgroup of S₁₄ and char(k)=7, then the fixed subfield k(x₁,⋯,x₁₄)^G is rational (i.e., purely transcendental) over k. In proving the above theorem, we rely on the Kuniyoshi–Gaschütz Theorem or some ideas in its proof.