The Rationality Problem for Three- and Four-Dimensional permutational Group Actions

Assume that K is a field, containing the full group of 4th roots of unity μ4, and char K = 2, 3. Let G be a finite non-abelian subgroup of GLn(K) for n = 3 or n = 4. The group G induces an action on K(x1, . . . , xn), the rational function field of n variables over K. Consider groups represented by matrices such that in each row and column there is exactly one element from μ4 and all other elements are 0. With the aid of GAP [3] we find that there are precisely 230 such non-abelian groups in SL4(K) and 33 in GL3(K), up to conjugacy. We show that the fixed subfield K(x1, . . . , xn) is rational (i.e. purely transcendental) over K for every such group G. We also give a positive answer to the Noether’s problem for several families of groups of order m = 2a · 3b, where a ≥ 2 and b = 0, 1.