[[Examples of groups]]
# Projective semilinear group

Let $V$ be a vector space over $\mathbb{K}$.
The **projective semilinear group** $\mathrm{P\Gamma L}(V)$ is the [[semidirect product]] #m/def/group 
$$
\begin{align*}
\mathrm{P\Gamma L}(V) = \Aut(\mathbb{K}) \ltimes  \mathrm{PGL}(V)
\end{align*}
$$
where $\mathrm{PGL}(V)$ is the [[projective general linear group]] acted on by $\sigma \in \Aut(\mathbb{K})$ coördinatewise, so $\sigma [A] \sigma^{-1} [\mathbf{x}]  = [\sigma(A)] [\mathbf{x}]$. By the [[Fundamental theorem of projective geometry]] this forms the [[Collineätion|collineätion group]] of the [[projectivization]] $\mathrm{P}(V)$.


#
---
#state/develop | #lang/en | #SemBr