Rational invariants for subgroups of S5 and S7

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(2), ... , x(n)) via k-automorphisms defined by sigma.x(i) = x(sigma(i)) for any sigma is an element of G, any 1 <= i <= n. Theorem. If n <= 5, then the fixed field k(x1, ... , x(n))(G) is purely transcendental over k. We will show that C(x(1), ... , X-7)(G) is also purely transcendental over C if G is any transitive subgroups of S-7 other than A(7); a similar result is valid for solvable transitive subgroups of S-11. (C) 2014 Elsevier Inc. All rights reserved.