Rationality problem for some meta-abelian groups
Let \(k\) be any field and let \(G\) be a finite group. Noether's problem asks whether \(k(V)^G\) is rational (= purely transcendental) over \(K\), where \(G\rightarrow \text{GL}(V)\) is a faithful finite-dimensional linear representation of \(G\) over \(k\). In particular when the representation is the regular representation of \(G\), one defines an action of \(G\) on the rational function field \(K(x_g: g\in G)\) by \(k\)-automorphisms where \(g: x_g = x_{gh}\) for any \(g, h\in G\). Denote by \(k(G) = k(x_g: g\in G)^G\), the fixed subfield. Noether's problem asks whether \(k(G)\) is rational over \(k\). In this paper the author proved the following Theorem. Assume that (i) \(G\) contains an abelian normal subgroup \(H\) so that \(G/H\) is cyclic of order \(n\), (ii) \(\mathbb{Z}(\zeta_n)\) is a unique factorization domain, and (iii)\(\zeta_e \in k\) where \(e\) is the exponent of \(G\), i.e., \(e = lcm\left\{\text{ord}(g): g\in G\right\}\). If \(G\rightarrow \text{GL}(V)\) is any finite-dimensional linear representation of \(G\) over \(k\), then \(k(V)^G\) is rational (= purely transcendental) over \(k\).
scientific article; zbMATH DE number 5611753