Noether ’ s problem for some groups of order 16

Noether’s problem then asks whether K(G) is rational (= purely transcendental) over K. Noether’s problem is closely related to the inverse Galois problem. The main results about Noether’s problem for abelian groups can be found in the survey article [Sw]. More recently, Noether’s problem for nonabelian p-groups was investigated in [CHK, CHPK, HK2, Ka1, Ka2]. Let n ≥ 2 be an arbitrary natural number. In this paper we will concentrate on certain meta-abelian groups of orders 8n and 16n with two or three generators over a field K which contains a primitive 4nth or 2nth root of unity. Let G be a non-abelian group of order 8n, having a cyclic subgroup of order 4n. Then G is generated by two elements σ and τ such that σ4n = 1, τ2 = σa and τσ = σrτ , where a, r ∈ Z are subject to some restrictions. For example, r must be a solution to the congruence