Editing General linear group (section)

== Related groups and monoids ==

=== Projective linear group ===
{{main article|Projective linear group}}
The [[projective linear group]] <math>\operatorname{PGL}(n,F)</math> and the [[projective special linear group]] <math>\operatorname{PSL}(n,F)</math> are the [[quotient group|quotients]] of <math>\operatorname{GL}(n,F)</math> and <math>\operatorname{SL}(n,F)</math> by their [[Group center|centers]] (which consist of the multiples of the identity matrix therein); they are the induced [[Group action (mathematics)|action]] on the associated [[projective space]].

=== 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>.

=== 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]].

=== Full linear monoid ===
The Full Linear Monoid, derived upon removal of the determinant's non-zero restriction, forms an algebraic structure akin to a monoid, often referred to as the full linear monoid or occasionally as the full linear semigroup or general linear monoid. Notably, it constitutes a regular semigroup.{{expand section|basic properties|date=April 2015}}
If one removes the restriction of the determinant being non-zero, the resulting algebraic structure is a [[monoid]], usually called the '''full linear monoid''',<ref name="Okniński1998">{{cite book|author=Jan Okniński|title=Semigroups of Matrices|year=1998|publisher=World Scientific|isbn=978-981-02-3445-4|at=Chapter 2: Full linear monoid}}</ref><ref name="Meakin">{{cite book|editor=C. M. Campbell|title=Groups St Andrews 2005|year=2007|publisher=Cambridge University Press|isbn=978-0-521-69470-4|page=471|chapter=Groups and Semigroups: Connections and contrast|author=Meakin}}</ref><ref name="RhodesSteinberg2009">{{cite book|author1=John Rhodes|author2=Benjamin Steinberg|title=The q-theory of Finite Semigroups|year=2009|publisher=Springer Science & Business Media|isbn=978-0-387-09781-7|page=306}}</ref> but occasionally also ''full linear semigroup'',<ref name="JespersOkniski2007">{{cite book|author1=Eric Jespers|author2=Jan Okniski|title=Noetherian Semigroup Algebras|year=2007|publisher=Springer Science & Business Media|isbn=978-1-4020-5810-3|at=2.3: Full linear semigroup}}</ref> ''general linear monoid''<ref name="Geck2013">{{cite book|author=Meinolf Geck|title=An Introduction to Algebraic Geometry and Algebraic Groups|year=2013|publisher=Oxford University Press|isbn=978-0-19-967616-3|page=132}}</ref><ref name="CanLi2014">{{cite book|author1=Mahir Bilen Can|author2=Zhenheng Li|author3=Benjamin Steinberg|author4=Qiang Wang|title=Algebraic Monoids, Group Embeddings, and Algebraic Combinatorics|year=2014|publisher=Springer|isbn=978-1-4939-0938-4|page=142}}</ref> etc. It is actually a [[regular semigroup]].<ref name="Meakin"/>