Editing Projective linear group (section)

== Low dimensions ==
The projective linear group is mostly studied for {{nowrap|''n'' ≥ 2}}, though it can be defined for low dimensions.

For {{nowrap|1=''n'' = 0}} (or in fact {{nowrap|''n'' < 0}}) the projective space of ''K''<sup>0</sup> is empty, as there are no 1-dimensional subspaces of a 0-dimensional space. Thus, {{nowrap|PGL(0, ''K'')}} is the trivial group, consisting of the unique empty map from the [[empty set]] to itself. Further, the action of scalars on a 0-dimensional space is trivial, so the map {{nowrap|''K''<sup>×</sup> → GL(0, ''K'')}} is trivial, rather than an inclusion as it is in higher dimensions.

For {{nowrap|1=''n'' = 1}}, the projective space of ''K''<sup>1</sup> is a single point, as there is a single 1-dimensional subspace. Thus, {{nowrap|PGL(1, ''K'')}} is the trivial group, consisting of the unique map from a [[singleton set]] to itself. Further, the general linear group of a 1-dimensional space is exactly the scalars, so the map {{nowrap|''K''<sup>×</sup> {{overset|lh=0.5|~|→}} GL(1, ''K'')}} is an isomorphism, corresponding to {{nowrap|1=PGL(1, ''K'') := GL(1, ''K''){{hsp}}/{{hsp}}''K''<sup>×</sup> ≅ {{mset|1}}}} being trivial.

For {{nowrap|1=''n'' = 2}}, {{nowrap|PGL(2, ''K'')}} is non-trivial, but is unusual in that it is 3-transitive, unlike higher dimensions when it is only 2-transitive.