Editing Projective linear group (section)

==== Action on projective line ====
Some of the above maps can be seen directly in terms of the action of PSL and PGL on the associated projective line: {{nowrap|PGL(''n'', ''q'')}} acts on the projective space '''P'''<sup>''n''−1</sup>(''q''), which has {{nowrap|(''q''<sup>''n''</sup> − 1)/(''q'' − 1)}} points, and this yields a map from the projective linear group to the symmetric group on {{nowrap|(''q''<sup>''n''</sup> − 1)/(''q'' − 1)}} points. For {{nowrap|1=''n'' = 2}}, this is the projective line '''P'''<sup>1</sup>(''q'') which has {{nowrap|1=(''q''<sup>2</sup> − 1)/(''q'' − 1) = ''q'' + 1}} points, so there is a map {{nowrap|PGL(2, ''q'') → S<sub>''q''+1</sub>}}.

To understand these maps, it is useful to recall these facts:
* The order of {{nowrap|PGL(2, ''q'')}} is
*: (''q''<sup>2</sup> − 1)(''q''<sup>2</sup> − ''q'')/(''q'' − 1) = ''q''<sup>3</sup> − ''q'' = (''q'' − 1)''q''(''q'' + 1);
: the order of {{nowrap|PSL(2, ''q'')}} either equals this (if the characteristic is 2), or is half this (if the characteristic is not 2).
* The action of the projective linear group on the projective line is sharply 3-transitive ([[faithful group action|faithful]] and 3-[[transitive group action|transitive]]), so the map is one-to-one and has image a 3-transitive subgroup.
Thus the image is a 3-transitive subgroup of known order, which allows it to be identified. This yields the following maps:
* {{nowrap|1=PSL(2, 2) = PGL(2, 2) → S<sub>3</sub>}}, of order 6, which is an isomorphism.
** The inverse map (a projective [[Dihedral group of order 6#Representation theory|representation of S<sub>3</sub>]]) can be realized by the [[anharmonic group]], and more generally yields an embedding {{nowrap|S<sub>3</sub> → PGL(2, ''q'')}} for all fields.
* {{nowrap|PSL(2, 3) < PGL(2, 3) → S<sub>4</sub>}}, of orders 12 and 24, the latter of which is an isomorphism, with {{nowrap|PSL(2, 3)}} being the alternating group.
** The anharmonic group gives a partial map in the opposite direction, mapping {{nowrap|S<sub>3</sub> → PGL(2, 3)}} as the stabilizer of the point −1.
* {{nowrap|1=PSL(2, 4) = PGL(2, 4) → S<sub>5</sub>}}, of order 60, yielding the alternating group A<sub>5</sub>.
* {{nowrap|PSL(2, 5) < PGL(2, 5) → S<sub>6</sub>}}, of orders 60 and 120, which yields an embedding of S<sub>5</sub> (respectively, A<sub>5</sub>) as a ''transitive'' subgroup of S<sub>6</sub> (respectively, A<sub>6</sub>). This is an example of an [[Automorphisms of the symmetric and alternating groups#Exotic map|exotic map {{nowrap|S<sub>5</sub> → S<sub>6</sub>}}]], and can be used to construct the [[Automorphisms of the symmetric and alternating groups#exceptional outer automorphism|exceptional outer automorphism of S<sub>6</sub>]].<ref>{{citation|title=Small finite sets |work=Secret Blogging Seminar] |date=2007-10-27 |first=Scott |last=Carnahan | url=http://sbseminar.wordpress.com/2007/10/27/small-finite-sets/ |postscript=, notes on a talk by [[Jean-Pierre Serre]].}}</ref> Note that the isomorphism {{nowrap|PGL(2, 5) ≅ S<sub>5</sub>}} is not transparent from this presentation: there is no particularly natural set of 5 elements on which {{nowrap|PGL(2, 5)}} acts.