Editing General linear group (section)

== Over finite fields ==
[[File:Symmetric group 3; Cayley table; GL(2,2).svg|thumb|[[Cayley table]] of {{nowrap|GL(2, 2)}}, which is isomorphic to [[Dihedral group of order 6|S<sub>3</sub>]].]]
If <math>F</math> is a [[finite field]] with <math>q</math> elements, then we sometimes write <math>\operatorname{GL}(n,q)</math> instead of <math>\operatorname{GL}(n,F)</math>. When ''p'' is prime, <math>\operatorname{GL}(n,p)</math> is the [[outer automorphism group]]  of the group <math>\Z_p^n</math>, and also the [[automorphism]] group, because <math>\Z_p^n</math> is abelian, so the [[inner automorphism group]] is trivial.

The order of <math>\operatorname{GL}(n,q)</math> is: 
: <math>\prod_{k=0}^{n-1}(q^n-q^k)=(q^n - 1)(q^n - q)(q^n - q^2)\ \cdots\ (q^n - q^{n-1}).</math>

This can be shown by counting the possible columns of the matrix: the first column can be anything but the zero vector; the second column can be anything but the multiples of the first column; and in general, the <math>k</math>th column can be any vector not in the [[linear span]] of the first <math>k-1</math> columns. In [[q-analog|''q''-analog]] notation, this is <math>[n]_q!(q-1)^n q^{n \choose 2}</math>.

For example, {{nowrap|GL(3, 2)}} has order {{nowrap|1=(8 − 1)(8 − 2)(8 − 4) = 168}}. It is the automorphism group of the [[Fano plane]] and of the group <math>\Z_2^3</math>. This group is also isomorphic to {{nowrap|[[PSL(2,7)|PSL(2, 7)]]}}.

More generally, one can count points of [[Grassmannian]] over <math>F</math>: in other words the number of subspaces of a given dimension <math>k</math>. This requires only finding the order of the [[stabilizer (group theory)|stabilizer]] subgroup of one such subspace and dividing into the formula just given, by the [[orbit-stabilizer theorem]].

These formulas are connected to the [[Schubert decomposition]] of the Grassmannian, and are [[q-analog|''q''-analogs]] of the [[Betti number]]s of complex Grassmannians. This was one of the clues leading to the [[Weil conjectures]].

Note that in the limit <math>q\to 1</math> the order of <math>\operatorname{GL}(n,q)</math> goes to 0! &ndash; but under the correct procedure (dividing by <math>(q-1)^n</math>) we see that it is the order of the symmetric group (see Lorscheid's article). In the philosophy of the [[field with one element]], one thus interprets the [[symmetric group]] as the general linear group over the field with one element: <math>S_n\cong \operatorname{GL}(n,1)</math>.

=== History ===
The general linear group over a prime field, <math>\operatorname{GL}(\nu,p)</math>, was constructed and its order computed by [[Évariste Galois]] in 1832, in his last letter (to Chevalier) and second (of three) attached manuscripts, which he used in the context of studying the [[Galois group]] of the general equation of order <math>p^\nu</math>.<ref name="chevalier-letter">{{cite journal
 | last = Galois
 | first = Évariste
 | year = 1846
 | title = Lettre de Galois à M. Auguste Chevalier
 | journal = [[Journal de Mathématiques Pures et Appliquées]]
 | volume = XI
 | pages = 408&ndash;415
 | url = http://visualiseur.bnf.fr/ark:/12148/cb343487840/date1846
 | access-date = 2009-02-04
 | postscript =, GL(''ν'',''p'') discussed on p. 410.}}</ref>