Noether’s problem for abelian extensions of cyclicp-groups

Abstract

In loving memory of my dear mother Let K be a field and G a finite group. Let G act on the rational function field K (x(g) : g ∈ G) by K-automorphisms defined by g • x(h) = x(gh) for any g, h ∈ G. Denote by K (G) the fixed field K (x(g) : g ∈ G) G. Noether's problem then asks whether K (G) is rational (i.e., purely transcendental) over K. The first main result of this article is that K (G) is rational over K for a certain class of p-groups having an abelian subgroup of index p. The second main result is that K (G) is rational over K for any group of order p 5 or p 6 (where p is an odd prime) having an abelian normal subgroup such that its quotient group is cyclic. (In both theorems we assume that if char K = p then K contains a primitive p e-th root of unity, where p e is the exponent of G.

Noether's problem for groups of order 243

“…We remark that Michailov shows that, if G is a group of order p 5 or p 6 such that G contains an abelian normal subgroup with cyclic quotient group, then k(G) is k-rational if k contains enough roots of unity [29,Theorem 1.9]. However, we will point out that the proof of [29, Theorem 1.9] contains a flaw.…”

“…However, we will point out that the proof of [29, Theorem 1.9] contains a flaw. Michailov's proof [29, Section 5, the first and second paragraphs] relies on [29,Theorem 1.8]. In applying this theorem, the condition that G (p) = {1} should be fulfilled.…”

“…For groups G = G(i), (28 ≤ i ≤ 30) which belong to Φ 10 , we take the following 9-dimensional faithful representations which are induced from a linear character on f 3 , f 4 , f 5 where (2) for G (29):…”

“…Let G be a group of order 243 which belongs to Φ 10 , i.e. G = G(i), ( For G = G (29) and G(30), we follow the same way to G (28) and take the same y ij 's. Define …”

“…Note that, for p = 3, the isoclinism family Φ 10 consists of the groups Φ 10 (2111)a r (where r = 0, 1) and Φ 10 (1 5 ) [21, page 621], which are just the groups G(243, 29), G(243, 30) and G(243, 28) in GAP code numbers respectively.…”