Editing General linear group (section)

== Special linear group ==
{{main article|Special linear group}}

The ''special linear group'', <math>\operatorname{SL}(n,F)</math>, is the group of all matrices with [[determinant]] 1. These matrices are special in that they lie on a [[Algebraic variety|subvariety]]: they satisfy a polynomial equation (as the determinant is a polynomial in the entries). Matrices of this type form a group as the determinant of the product of two matrices is the product of the determinants of each matrix.

If we write <math>F^\times</math> for the [[multiplicative group]] of <math>F</math> (that is, <math>F</math> excluding 0), then the determinant is a [[group homomorphism]]
:<math>\det : \operatorname{GL}(n,F) \to F^\times</math>
that is surjective and its [[kernel (algebra)|kernel]] is the special linear group. Thus, <math>\operatorname{SL}(n,F)</math> is a [[normal subgroup]] of <math>\operatorname{GL}(n,F)</math>, and by the [[first isomorphism theorem]], <math>\operatorname{GL}(n,F)/\operatorname{SL}(n,F)</math> is [[isomorphic]] to <math>F^\times</math>. In fact, <math>\operatorname{GL}(n,F)</math> can be written as a [[semidirect product]]:
:<math>\operatorname{GL}(n,F)=\operatorname{SL}(n,F)\rtimes F^\times </math>.

The special linear group is also the [[derived group]] (also known as commutator subgroup) of <math>\operatorname{GL}(n,F)</math> (for a field or a [[division ring]] <math>F</math>), provided that <math>n \neq 2</math> or <math>F</math> is not the [[finite field|field with two elements]].<ref>{{citation|author=Suprunenko|first=D.A.|title=Matrix groups|publisher=American Mathematical Society|year=1976|series=Translations of Mathematical Monographs}}, Theorem II.9.4</ref>

When <math>F</math> is <math>\R</math> or <math>\C</math>, <math>\operatorname{SL}(n,F)</math> is a [[Lie subgroup]] of <math>\operatorname{GL}(n,F)</math> of dimension <math>n^2-1</math>. The [[Lie algebra]] of <math>\operatorname{SL}(n,F)</math> consists of all <math>n\times n</math> matrices over <math>F</math> with vanishing [[trace (matrix)|trace]]. The Lie bracket is given by the [[commutator]].

The special linear group <math>\operatorname{SL}(n,\R)</math> can be characterized as the group of ''[[volume]] and [[orientation-preserving]]'' linear transformations of <math>\R^n</math>.

The group <math>\operatorname{SL}(n,\C)</math> is simply connected, while <math>\operatorname{SL}(n,\R)</math> is not. <math>\operatorname{SL}(n,\R)</math> has the same fundamental group as <math>\operatorname{GL}^+(n,\R)</math>, that is, <math>\Z</math> for <math>n=2</math> and <math>\Z_2</math> for <math>n>2</math>.