Generic Picard–Vessiot Extensions for Nonconnected Groups

Let K be a differential field with algebraically closed field of constants 𝒞 and G a linear algebraic group over 𝒞. We provide a characterization of the K-irreducible G-torsors for nonconnected groups G in terms of the first Galois cohomology H¹(K, G) and use it to construct Picard–Vessiot extensions which correspond to nontrivial torsors for the infinite quaternion group, the infinite multiplicative and additive dihedral groups and the orthogonal groups. The extensions so constructed are generic for those groups.