Editing Projective linear group (section)

== Name ==
The name comes from [[projective geometry]], where the projective group acting on [[homogeneous coordinates]] (''x''<sub>0</sub> : ''x''<sub>1</sub> : ... : ''x<sub>n</sub>'') is the underlying group of the geometry.<ref group="note">This is therefore {{nowrap|PGL(''n'' + 1, ''F'')}} for [[projective space]] of dimension ''n''</ref> Stated differently, the natural [[Group action (mathematics)|action]] of GL(''V'') on ''V'' descends to an action of PGL(''V'') on the projective space ''P''(''V'').

The projective linear groups therefore generalise the case {{nowrap|PGL(2, '''C''')}} of [[Möbius transformation]]s (sometimes called the [[Möbius group]]), which acts on the [[projective line]].

Note that unlike the general linear group, which is generally defined axiomatically as "invertible functions preserving the linear (vector space) structure", the projective linear group is defined ''constructively,'' as a quotient of the general linear group of the associated vector space, rather than axiomatically as "invertible functions preserving the projective linear structure". This is reflected in the notation: {{nowrap|PGL(''n'', ''F'')}} is the group associated to {{nowrap|GL(''n'', ''F'')}}, and is the projective linear group of {{nowrap|(''n'' − 1)}}-dimensional projective space, not ''n''-dimensional projective space.

=== Collineations ===
{{main|Collineation}}
A related group is the [[collineation group]], which is defined axiomatically. A collineation is an invertible (or more generally one-to-one) map which sends [[collinear points]] to collinear points. One can [[Projective space#Axioms for projective space|define a projective space axiomatically]] in terms of an [[incidence structure]] (a set of points ''P'', lines ''L'', and an [[incidence relation]] ''I'' specifying which points lie on which lines) satisfying certain axioms – an automorphism of a projective space thus defined then being an automorphism ''f'' of the set of points and an automorphism ''g'' of the set of lines, preserving the incidence relation,<ref group="note">"Preserving the incidence relation" means that if point ''p'' is on line ''l'' then ''f''(''p'') is in ''g''(''l''); formally, if {{nowrap|(''p'', ''l'') ∈ ''I''}} then {{nowrap|(''f''(''p''), ''g''(''l'')) ∈ ''I''}}.</ref> which is exactly a collineation of a space to itself. Projective linear transforms are collineations (planes in a vector space correspond to lines in the associated projective space, and linear transforms map planes to planes, so projective linear transforms map lines to lines), but in general not all collineations are projective linear transforms – PGL is in general a proper subgroup of the collineation group.

Specifically, for {{nowrap|1=''n'' = 2}} (a projective line), all points are collinear, so the collineation group is exactly the [[symmetric group]] of the points of the projective line, and except for '''F'''<sub>2</sub> and '''F'''<sub>3</sub> (where PGL is the full symmetric group), PGL is a proper subgroup of the full symmetric group on these points.

For {{nowrap|''n'' ≥ 3}}, the collineation group is the [[projective semilinear group]], PΓL – this is PGL, twisted by [[field automorphism]]s; formally, {{nowrap|PΓL ≅ PGL ⋊ Gal(''K''{{hsp}}/{{hsp}}''k'')}}, where ''k'' is the [[prime field]] for ''K''; this is the [[fundamental theorem of projective geometry]]. Thus for ''K'' a prime field ('''F'''<sub>''p''</sub> or '''Q'''), we have {{nowrap|1=PGL = PΓL}}, but for ''K'' a field with non-trivial Galois automorphisms (such as '''F'''<sub>''p''<sup>''n''</sup></sub> for {{nowrap|''n'' ≥ 2}} or '''C'''), the projective linear group is a proper subgroup of the collineation group, which can be thought of as "transforms preserving a projective ''semi''-linear structure". Correspondingly, the quotient group {{nowrap|1=PΓL{{hsp}}/{{hsp}}PGL = Gal(''K''{{hsp}}/{{hsp}}''k'')}} corresponds to "choices of linear structure", with the identity (base point) being the existing linear structure.

One may also define collineation groups for axiomatically defined projective spaces, where there is no natural notion of a projective ''linear'' transform. However, with the exception of the [[non-Desarguesian plane]]s, all projective spaces are the projectivization of a linear space over a [[division ring]] though, as noted above, there are multiple choices of linear structure, namely a [[torsor]] over Gal(''K''{{hsp}}/{{hsp}}''k'') (for {{nowrap|''n'' ≥ 3}}).