Degree three unramified cohomology groups and Noether's problem for groups of order 243

Abstract

Let k be a field and G be a finite group acting on the rational function field k(xg : g ∈ G) by k-automorphisms defined as h(xg) = x hg for any g, h ∈ G. We denote the fixed field k(xg : g ∈ G) G by k(G). Noether's problem asks whether k(G) is rational (= purely transcendental) over k. It is well-known that if (G) is stably rational over , then all the unramified cohomology groups H i nr ( (G), É/ ) = 0 for i ≥ 2. Hoshi, Kang and Kunyavskii [HKK] showed that, for a p-group of order p 5 (p: an odd prime number), H 2 nr ( (G), É/ ) = 0 if and only if G belongs to the isoclinism family Φ 10 . When p is an odd prime number, Peyre [Pe3] and Hoshi, Kang and Yamasaki [HKY1] exhibit some p-groups G which are of the form of a central extension of certain elementary abelian p-group by another one with H 2 nr ( (G), É/ ) = 0 and H 3 nr ( (G), É/ ) = 0. However, it is difficult to tell whether H 3 nr ( (G), É/ ) is non-trivial if G is an arbitrary finite group. In this paper, we are able to determine H 3 nr ( (G), É/ ) where G is any group of order p 5 with p = 3, 5, 7. Theorem 1. Let G be a group of order 3 5 . Then H 3 nr ( (G), É/ ) = 0 if and only if G belongs to the isoclinism family Φ 7 . Theorem 2. If G is a group of order 3 5 , then the fixed field (G) is rational if and only if G does not belong to the isoclinism families Φ 7 and Φ 10 . Theorem 3. Let G be a group of order 5 5 or 7 5 . Then H 3 nr ( (G), É/ ) = 0 if and only ifG belongs to the isoclinism families Φ 6 , Φ 7 or Φ 10 . Theorem 4. If G is the alternating group An, the Mathieu group M 11 , M 12 , the Janko group J 1 or the group P SL 2 ( q ), SL 2 ( q ), P GL 2 ( q ) (where q is a prime power), then H d nr ( (G), É/ ) = 0 for any d ≥ 2. Besides the degree three unramified cohomology groups, we compute also the stable cohomology groups.