On Picard–Vessiot extensions with group PGL3

On Picard–Vessiot extensions with group PGL3

Let F be a differential field of characteristic zero with algebraically closed field of constants. We provide an explicit description of the twisted Lie algebras of PGL3-equivariant derivations on the coordinate rings of F -irreducible PGL3-torsors in terms of nine-dimensional central simple algebras over F . We use this to construct a Picard–Vessiot extension which is the function field of a n...

Group Actions and Group Extensions

In this paper we study finite group extensions represented by special cohomology classes. As an application, we obtain some restrictions on finite groups which can act freely on a product of spheres or on a product of real projective spaces. In particular, we prove that if (Z/p)r acts freely on (S1)k , then r ≤ k.

On Linearization of Lie Group Extensions

Cohomology of Group Extensions

Introduction. Let G be a group, K an invariant subgroup of G. The purpose of this paper is to investigate the relations between the cohomology groups of G, K, and G/K. As in the case of fibre spaces, it turns out that such relations can be expressed by a spectral sequence whose term E2 is HiG/K, HiK)) and whose term Em is the graduated group associated with i7(G). This problem was first studied...

Group Extensions and Automorphism Group Rings

We use extensions to study the semi-simple quotient of the group ring FpAut(P ) of a finite p-group P . For an extension E : N → P → Q, our results involve relations between Aut(N), Aut(P ), Aut(Q) and the extension class [E] ∈ H(Q,ZN). One novel feature is the use of the intersection orbit group Ω([E]), defined as the intersection of the orbits Aut(N) · [E] and Aut(Q) · [E] in H(Q,ZN). This gr...