The Brauer-Clifford group and rational forms of finite groups.
Let \(k\) be a perfect field and let \(G\) be a finite subgroup of \(\mathrm{GL}_n(\overline k)\). The aim of this paper is to study the following question: Is it possible to find a subgroup of \(\mathrm{GL}_n(\overline k)\), conjugate to \(G\), with a set of fundamental polynomial invariants whose coefficients lie in the field \(k\)? Equivalently, is it possible to find a subgroup of \(\mathrm{GL}_n(\overline k)\), conjugate to \(G\), and defined over \(k\) when viewed as an algebraic group? (That this is equivalent is Theorem 2.1 in the paper.) The Brauer-Clifford group introduced by Turull is a variant of the Brauer group, taking into account the action of a Galois group \(U\) on ``central simple \(U\)-algebras''. Using the Brauer-Clifford group the question above 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 is introduced. It can be identified with a subdirect product of the Brauer group and a second cohomology group. The main question is answered completely if \(G\) is an absolutely irreducible subgroup of \(\mathrm{GL}_n(\overline{\mathbb Q})\) and isomorphic to \(\mathrm{PSL}_2(\mathbb F_q)\). Furthermore, if \(k\) is the real field or a finite field then a satisfactory treatment is possible for any finite group.
scientific article; zbMATH DE number 5994289