The rationality problem for finite subgroups of GL 4(Q)
Journal
Journal of Algebra
Journal Volume
368
Pages
53-69
Date Issued
2012
Author(s)
Abstract
Let G be a finite subgroup of GL 4(Q). The group G induces an action on Q(x 1,x 2,x 3,x 4), the rational function field of four variables over Q. Theorem. The fixed subfield Q(x 1,x 2,x 3,x 4)G:={f∈Q(x 1,x 2,x 3,x 4):σ{dot operator}f=ffor anyσ∈G} is rational (i.e. purely transcendental) over Q, except for two groups which are images of faithful representations of C 8 and C 3⋊C 8 into GL 4(Q) (both fixed fields for these two exceptional cases are not rational over Q). There are precisely 227 such groups in GL 4(Q) up to conjugation; the answers to the rationality problem for most of them were proved by Kitayama and Yamasaki (2009) [KY] except for four cases. We solve these four cases left unsettled by Kitayama and Yamasaki; thus the whole problem is solved completely. © 2012 Elsevier Inc.
Type
journal article