Editing General linear group (section)

=== General semilinear group ===
{{main article|General semilinear group}}
The [[general semilinear group]] <math>\operatorname{\Gamma L}(n,F)</math> is the group of all invertible [[semilinear transformation]]s, and contains <math>\operatorname{GL}(n,F)</math>. A semilinear transformation is a transformation which is linear “up to a twist”, meaning “up to a [[field automorphism]] under scalar multiplication”. It can be written as a semidirect product:
:<math>\operatorname{\Gamma L}(n,F)=\operatorname{Gal}(F)\ltimes \operatorname{GL}(n,F)</math>
where <math>\operatorname{Gal}(F)</math> is the [[Galois group]] of <math>F</math> (over its [[prime field]]), which acts on <math>\operatorname{GL}(n,F)</math> by the Galois action on the entries.

The main interest of <math>\operatorname{\Gamma L}(n,F)</math> is that the associated [[projective semilinear group]] <math>\operatorname{P\Gamma L}(n,F)</math>, which contains <math>\operatorname{PGL}(n,F)</math>, is the [[collineation group]] of [[projective space]], for <math>n>2</math>, and thus semilinear maps are of interest in [[projective geometry]].