Rationality problem for some meta-abelian groups
type: Item
instance of: scholarly article
title: Rationality problem for some meta-abelian groups
zbMATH Open document ID: 1181.13004
published in: Journal of Algebra
publication date: 2009-10-09T00:00:00Z
review text: 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 \(k\).
reviewed by: Jebrel M. Habeb
Mathematics Subject Classification ID: 13A50
Mathematics Subject Classification ID: 16W22
zbMATH DE Number: 5611753
zbMATH Keywords: Noether's problem
zbMATH Keywords: rationality problem
zbMATH Keywords: multiplicative group actions
zbMATH Keywords: meta-abelian group actions
author: Ming-Chang Kang
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2009.04.030
OpenAlex ID: W2013855865
cites work: Rationality of some projective linear actions
cites work: THE BRAUER GROUP OF QUOTIENT SPACES BY LINEAR GROUP ACTIONS
cites work: Noether's problem for groups of order 32
cites work: Rationality of \(P\)-group actions
cites work: On the Invariants of Finite Groups Having an Abelian Normal Subgroup of Prime Index
cites work: Rational invariants of meta-abelian groups of linear automorphisms
cites work: Noether's problem for metacyclic \(p\)-groups
cites work: Rational functions invariant under a finite Abelian group
cites work: Q4101931
cites work: Noether's problem over an algebraically closed field
cites work: Lüroth's problem
cites work: Q3324057
DOI: 10.1016/J.JALGEBRA.2009.04.030
description: scientific article; zbMATH DE number 5611753