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Special linear group
 
 
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In mathematicsMathematics

Mathematics is the discipline that deals with concepts such as quantity, structure, space and change....
, the special linear group of degree n over a fieldField (mathematics)

In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and ...
 F is the set of n×n matricesMatrix (mathematics) Summary

In mathematics, a matrix is a rectangular table of numbers or, more generally, a table consisting of abstract quantities tha...
 with determinantDeterminant

In algebra, a determinant is a function depending on n that associates a scalar, det, to every n'n square matrix,...
 1, with the group operations of ordinary matrix multiplicationMatrix multiplication

This article gives an overview of the various ways to multiply matrices. ...
 and matrix inversion.
This is the normal subgroupNormal subgroup

In mathematics, more specifically in abstract algebra, a normal subgroup is a special kind of subgroup....
 of the general linear groupGeneral linear group

In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with ...
, given by the kernelKernel (algebra)

In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measure...
 of the determinant

where we write F× for the multiplicative group of F (that is, excluding 0).

Geometric interpretation

The special linear group SL(n, R) can be characterized as the group of volumeVolume

'Volume', also called capacity, is a quantification of how much space a certain region occupies....
 and orientationOrientation (mathematics)

In mathematics, an orientation on a real vector space is a choice of which ordered bases are "positively" oriented and which...
 preserving linear transformations of Rn; this corresponds to the interpretation of the determinant as measuring change in volume and orientation.

Lie subgroup

When F is R or C, SL(n) is a Lie subgroupLie subgroup

In mathematics, a subgroup H of a Lie group G is a Lie subgroup if it is also a submanifold of G....
 of GL(n) of dimension n2 − 1. The Lie algebraLie algebra

In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups a...
 of SL(n) consists of all n×n matrices over F with vanishing trace. The Lie bracket is given by the commutatorFacts About Commutator

In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative...
.

Topology

The group SL(n, C) is simply connected while SL(n, R) is not. SL(n, R) has the same fundamental group as GL+(n, R), that is, Z for n = 2 and Z2 for n > 2.

Relations to other subgroups of GL(n,A)

Two related subgroups, which in some cases coincide with SL, and in other cases are accidentally conflated with SL, are the commutator subgroupCommutator subgroup

In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the s...
 of GL, and the group generated by transvectionTransvection

Transvection can refer to one of the following items:...
s. These are both subgroups of SL (transvections have determinant 1, and det is a map to an abelian group, so ), but in general do not coincide with it.

The group generated by transvections is denoted
(for elementary matrices) or . By the second Steinberg relation, for , transvections are commutators, so
for , .
For , transvections need not be commutators (of 2×2 matrices), as seen for example when is the field of two elements, then .

In some circumstances these coincide: the special linear group over a field or the integers is generated by transvections, and the stable special linear group over a Dedekind domainDedekind domain

In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is a Noetherian integral domain...
 is generated by transvections. For more general rings the stable difference is measured by the special Whitehead group , where and are the stable groupDirect limit of groups

In mathematics, a direct limit of groups is the direct limit of a direct system of groups....
s of the special linear group and elementary matrices.

Generators and relations

If working over a ring where SL is generated by transvections (such as a ring or the integers), one can give a presentationPresentation of a group

In mathematics, one method of defining a group is by a presentation....
 of SL using transvections with some relations. Transvections satisfy the Steinberg relations, but these are not sufficient: the resulting group is the Steinberg groupFacts About Steinberg group

In mathematics, Steinberg group means one of two distinct, though closely related, constructions of the mathematician Robert...
, which is not the special linear group, but rather the universal central extension of the commutator subgroup of GL.

A sufficient set of relations for for is given by two of the Steinberg relations, plus a third relation .
Let be the elementary matrix with 1's on the diagonal and in the position, and 0's elsewhere (and ). Then

are a complete set of relations for , .

Structure of GL(n,F)

The group splits over its determinant (we use as the group homomorphismFacts About Group homomorphism

In mathematics, given two groups and , a group homomorphism from to is a function h : G → H such that fo...
 needed from to ,
see semidirect productSemidirect product

In group theory, a semidirect product describes a particular way in which a group can be put together from two subgroups, on...
), and therefore GL(n, F) can be written as a semidirect productSemidirect product

In group theory, a semidirect product describes a particular way in which a group can be put together from two subgroups, on...
 of SL(n, F) by F×:
GL(n, F) = SL(n, F) ? F×