Editing General linear group (section)

=== Affine group ===
{{main article|Affine group}}

The [[affine group]] <math>\operatorname{Aff}(n,F)</math> is an [[group extension|extension]] of <math>\operatorname{GL}(n,F)</math> by the group of translations in <math>F^n</math>. It can be written as a [[semidirect product]]:
:<math>\operatorname{Aff}(n,F)=\operatorname{GL}(n,F)\ltimes F^n </math>
where <math>\operatorname{GL}(n,F)</math> acts on <math>F^n</math> in the natural manner. The affine group can be viewed as the group of all [[affine transformation]]s of the [[affine space]] underlying the vector space <math>F^n</math>.

One has analogous constructions for other subgroups of the general linear group: for instance, the [[special affine group]] is the subgroup defined by the semidirect product, <math>\operatorname{SL}(n,F)\ltimes F^n </math>, and the [[Poincaré group]] is the affine group associated to the [[Lorentz group]], <math>\operatorname{O}(1,3,F)\ltimes F^n </math>.