Noether Settings of Some Groups with an Unexpected Property

Abstract

Let p be an odd prime, and let G be the symmetric group on p letters. Let F be a field of characteristic zero containing a primitive p 2th root of 1. Let H be a p-Sylow subgroup of G, and let be a nontrivial ZH-module of order p 2, cyclic as an abelian group. Let be its character group. We show that the invariant subfields of the Noether settings of the groups and are stably rational over F. We also show that there is a set of quotient groups and subgroups of G′ and of G″, whose Noether settings have stably rational invariant subfields over F. The group G′ arose in the study of the center of the generic division algebra of degree p, and we exhibit a subgroup of G′ with the unexpected property that the invariant subfield of its Noether setting is stably isomorphic to this center.