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

Abstract 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 ( x 1 , ⋯ , 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 S 14 and char ( k ) = 7 , then the fixed subfield k ( x 1 , ⋯ , x 14 ) G is rational (i.e., purely transcendental) over k. In proving the above theorem, we rely on the Kuniyoshi–Gaschutz Theorem or some ideas in its proof.