Editing Projective linear group (section)

{{Short description|Construction in group theory}}
{{Redirect|Projective group}}
[[File:PSL-PGL.svg|thumb|491px|Relation between the projective special linear group PSL and the projective general linear group PGL; each row and column is a [[short exact sequence]]. The set (''F''<sup>*</sup>)<sup>''n''</sup> here is the set of ''n''th powers of the multiplicative group of ''F''.]]
{{Lie groups}}

In [[mathematics]], especially in the [[group theory|group theoretic]] area of [[algebra]], the '''projective linear group''' (also known as the '''projective general linear group''' or PGL) is the induced [[Group action (mathematics)|action]] of the [[general linear group]] of a [[vector space]] ''V'' on the associated [[projective space]] P(''V''). Explicitly, the projective linear group is the [[quotient group]]
: PGL(''V'') = GL(''V''){{hsp}}/{{hsp}}Z(''V'')
where GL(''V'') is the [[general linear group]] of ''V'' and Z(''V'') is the subgroup of all nonzero [[scalar transformation]]s of ''V''; these are quotiented out because they act [[trivial action|trivially]] on the projective space and they form the [[Kernel (algebra)|kernel]] of the action, and the notation "Z" reflects that the scalar transformations form the [[center of a group|center]] of the general linear group.

The '''projective special linear group''', PSL, is defined analogously, as the induced action of the [[special linear group]] on the associated projective space. Explicitly:
: PSL(''V'') = SL(''V''){{hsp}}/{{hsp}}SZ(''V'')
where SL(''V'') is the special linear group over ''V'' and SZ(''V'') is the subgroup of scalar transformations with unit [[determinant]]. Here SZ is the center of SL, and is naturally identified with the group of ''n''th [[roots of unity]] in ''F'' (where ''n'' is the [[Dimension (vector space)|dimension]] of ''V'' and ''F'' is the base [[Field (mathematics)|field]]).

PGL and PSL are some of the fundamental groups of study, part of the so-called [[classical groups]], and an element of PGL is called  '''projective linear transformation''', '''projective transformation''' or '''[[homography]]'''. If ''V'' is the ''n''-dimensional vector space over a field ''F'', namely {{nowrap|1 = ''V'' = ''F''<sup>''n''</sup>}}, the alternate notations {{nowrap|PGL(''n'', ''F'')}} and {{nowrap|PSL(''n'', ''F'')}} are also used.

Note that {{nowrap|PGL(''n'', ''F'')}} and {{nowrap|PSL(''n'', ''F'')}} are [[Group isomorphism|isomorphic]] [[if and only if]] every element of ''F'' has an ''n''th root in&nbsp;''F''.  As an example, note that {{nowrap|1 = PGL(2, '''C''') = PSL(2, '''C''')}}, but that {{nowrap|1 = PGL(2, '''R''') > PSL(2, '''R''')}};<ref>Gareth A. Jones and David Singerman. (1987)  Complex functions: an algebraic and geometric viewpoint.  Cambridge UP.  [https://books.google.com/books?id=jJhWM4vAyVMC&dq=psl+pgl&pg=PA20 Discussion of PSL and PGL on page 20 in google books]</ref> this corresponds to the real projective line being orientable, and the projective special linear group only being the orientation-preserving transformations.

PGL and PSL can also be defined over a [[Ring (mathematics)|ring]], with an important example being the [[modular group]], {{nowrap|1=PSL(2, '''Z''')}}.