Computing Galois cohomology and forms of linear algebraic groups

Abstract

We design and implement algorithms for computation with groups of Lietype. Algorithms for element arithmetic in the Steinberg presentation ofuntwisted groups of Lie type, and for conversion between this presentationand linear representations, were given in [12] (building on work of [15]and [26]). We extend this work to twisted groups, including groups thatare not quasisplit. A twisted group of Lie type is the group of rational points of a twistedform of a reductive linear algebraic group. These forms are classified byGalois cohomology. In order to compute the Galois cohomology, we develop amethod for computing the cohomology of a finitely presented group $\Gamma$on a finite group $A$. This method is of interest in its own right. Wethen extend this method to the Galois cohomology of reductive linearalgebraic groups. We give algorithms for computing the relative root system of a twistedgroup of Lie type, the root subgroups, and the root elements, as well asalgorithms for the computing of relations between root elements. As an application, we develop an algorithm for computing all twistedmaximal tori of a finite group of Lie type. The order of such a torus iscomputed as a polynomial in $q$, the order of the field $k$. We alsocompute the orders of the factors in a decomposition of the torus as adirect product of cyclic subgroups.