Vol. 270, No. 1, 2014

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Noether's problem for abelian extensions of cyclic $p$-groups

Ivo M. Michailov

Vol. 270 (2014), No. 1, 167–189
Abstract

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 p5 or p6 (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 charKp then K contains a primitive pe-th root of unity, where pe is the exponent of G.)

In loving memory of my dear mother

Keywords
Noether's problem, rationality problem, metabelian group actions
Mathematical Subject Classification 2010
Primary: 14E08, 14M20
Secondary: 13A50, 12F12
Milestones
Received: 18 January 2013
Accepted: 13 February 2013
Published: 2 August 2014
Authors
Ivo M. Michailov
Faculty of Mathematics and Informatics
Shumen University “Episkop Konstantin Preslavsky”
Universitetska Street 115
9700 Shumen
Bulgaria