Rational invariants for subgroups of S5 and S7

Let G be a subgroup of Sn, the symmetric group of degree n. For any field k, G acts naturally on the rational function field k(x1,x2,…,xn) via k-automorphisms defined by σ⋅xi=xσ(i) for any σ∈G, any 1≤i≤n. TheoremIf n≤5, then the fixed field k(x1,…,xn)Gis purely transcendental over k. We will show that C(x1,…,x7)G is also purely transcendental over C if G is any transitive subgroups of S7 other than A7; a similar result is valid for solvable transitive subgroups of S11.