Editing General linear group (section)

== Other subgroups ==

=== Diagonal subgroups ===

The set of all invertible [[diagonal matrix|diagonal matrices]] forms a subgroup of <math>\operatorname{GL}(n,F)</math> isomorphic to <math>(F^\times)^n</math>. In fields like <math>\R</math> and <math>\C</math>, these correspond to rescaling the space; the so-called dilations and contractions.

A '''scalar matrix''' is a diagonal matrix which is a constant times the [[identity matrix]]. The set of all nonzero scalar matrices forms a subgroup of <math>\operatorname{GL}(n,F)</math> isomorphic to <math>F^\times</math>. This group is the [[center of a group|center]] of <math>\operatorname{GL}(n,F)</math>. In particular, it is a normal, abelian subgroup.

The center of <math>\operatorname{SL}(n,F)</math> is simply the set of all scalar matrices with unit determinant, and is isomorphic to the group of <math>n</math>th [[roots of unity]] in the field <math>F</math>.

=== Classical groups ===

The so-called [[classical group]]s are subgroups of <math>\operatorname{GL}(V)</math> which preserve some sort of [[bilinear form]] on a vector space <math>V</math>. These include the
* '''[[orthogonal group]]''', <math>\operatorname{O}(V)</math>, which preserves a [[non-degenerate]] [[quadratic form]] on <math>V</math>,
* '''[[symplectic group]]''', <math>\operatorname{Sp}(V)</math>, which preserves a [[Symplectic vector space|symplectic form]] on <math>V</math> (a non-degenerate [[alternating form]]),
* '''[[unitary group]]''', <math>\operatorname{U}(V)</math>, which, when <math>F=\C</math>, preserves a non-degenerate [[hermitian form]] on <math>V</math>.
These groups provide important examples of Lie groups.