The Brauer–Clifford group and rational forms of finite groups

Abstract

Let k be a perfect field and G be a finite subgroup of GL n ( k ¯ ) . The aim of this paper is to study the following question: Is it possible to find a subgroup of GL n ( k ¯ ) , conjugate to G, with a set of fundamental polynomial invariants whose coefficients lie in the field k? Using the Brauer–Clifford group introduced by Turull this question is split into two natural parts. The first part is to recognize elements in the Brauer–Clifford group and the second part is to decide a field theoretic question involving Galois cohomology. For the first part a subgroup of the Brauer–Clifford group, which can be identified with a subdirect product of the Brauer group and a second cohomology group, is introduced. The main question is answered completely if G is an absolutely irreducible subgroup of GL n ( Q ¯ ) and isomorphic to PSL 2 ( F q ) . Furthermore, if k is the real field or a finite field then a satisfactory treatment is possible for any finite group.