Invariants of wreath products and subgroups of S₆

Let G be a subgroup of S₆, the symmetric group of degree 6. For any field κ, G acts naturally on the rational function field k(x₁,. . .x₆) via κ-automorphisms defined by ρ • x_i = x_ρ(i) for any ρ ∈ G and any 1 ≤i ≤ 6.We prove the following theorem. The fixed field κ(x₁,. . .x₆)^G is rational (i.e., purely transcendental) over κ, except possibly when G is isomorphic toPSL₂(F₅),PGL₂(F₅), orA₆.When G is isomorphic toPSL₂(F₅) or PGL₂(F₅), then C(x₁,. . .x₆)G is C-rational and κ(x₁,. . .x₆)^G is stably κ-rational for any field κ. The invariant theory of wreath products will be investigated also.