Projective semilinear group

Let $𝑉$ be a vector space over $𝕂$ . The projective semilinear group $P Γ L (𝑉)$ is the semidirect product group

P Γ L (𝑉) = A u t (𝕂) ⋉ P G L (𝑉)

where $P G L (𝑉)$ is the projective general linear group acted on by $𝜎 \in A u t (𝕂)$ coördinatewise, so $𝜎 [𝐴] 𝜎 - 1 [𝐱] = [𝜎 (𝐴)] [𝐱]$ . By the Fundamental theorem of projective geometry this forms the collineätion group of the projectivization $P (𝑉)$ .