projective special linear group


Let V be a vector spaceMathworldPlanetmath over a field F and let SL(V) be the special linear groupMathworldPlanetmath. Let Z be the center of SL(V). The projective special linear groupMathworldPlanetmath associated to V is the quotient groupMathworldPlanetmath SL(V)/Z and is usually denoted by PSL(V).

When V is a finite dimensional vector space over F (of dimensionPlanetmathPlanetmath n) then we write PSL(n,F) or PSLn(F). We also identify the linear transformations of V with n×n matrices, so PSL may be regarded as a quotient of the group of matrices SL(n,F) by its center.

Note: see the entry on projective spaceMathworldPlanetmath for the origin of the terminology.

Theorem 1.

The center Z of SL(n,F) is the group of all scalar matrices λId where λ is an nth root of unityMathworldPlanetmath in F.

In particular, for n=2, Z={±Id} and:


As a consequence of the previous theorem, we obtain:

Theorem 2.

For n3, PSL(n,F) is a simple groupMathworldPlanetmathPlanetmath. Furthermore, if F is a finite field then the groups


are all finite simple groups, except for n=2 and F=F2,F3.


Title projective special linear group
Canonical name ProjectiveSpecialLinearGroup
Date of creation 2013-03-22 15:09:46
Last modified on 2013-03-22 15:09:46
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 4
Author alozano (2414)
Entry type Definition
Classification msc 20G15
Synonym PSL
Related topic TheoremsOfSpecialLinearGroupOverAFiniteField