Editing Projective linear group (section)

== Properties ==
* PGL sends collinear points to collinear points (it preserves projective lines), but it is not the full [[collineation group]], which is instead either PΓL (for {{nowrap|''n'' > 2}}) or the full [[symmetric group]] for {{nowrap|1=''n'' = 2}} (the projective line).
* Every ([[biregular]]) algebraic automorphism of a projective space is projective linear. The [[birational automorphism]]s form a larger group, the [[Cremona group]].
* PGL acts faithfully on projective space: non-identity elements act non-trivially.{{br}} Concretely, the kernel of the action of GL on projective space is exactly the scalar maps, which are quotiented out in PGL.
* PGL acts [[2-transitive group|2-transitively]] on projective space.{{br}} This is because 2 distinct points in projective space correspond to 2 vectors that do not lie on a single linear space, and hence are [[linearly independent]], and GL acts transitively on ''k''-element sets of linearly independent vectors.
* {{nowrap|PGL(2, ''K'')}} acts sharply 3-transitively on the projective line.{{br}} Three arbitrary points are conventionally mapped to [0, 1], [1, 1], [1, 0]; in alternative notation, 0, 1, ∞. In fractional linear transformation notation, the function {{nowrap|{{sfrac|''x'' − ''a''|''x'' − ''c''}} ⋅ {{sfrac|''b'' − ''c''|''b'' − ''a''}}}} maps {{nowrap|''a'' ↦ 0}}, {{nowrap|''b'' ↦ 1}}, {{nowrap|''c'' ↦ ∞}}, and is the unique such map that does so. This is the [[cross-ratio]] {{nowrap|(''x'', ''b''; ''a'', ''c'')}} – see ''{{slink|Cross-ratio#Transformational approach}}'' for details.
* For {{nowrap|''n'' ≥ 3}}, {{nowrap|PGL(''n'', ''K'')}} does not act 3-transitively, because it must send 3 collinear points to 3 other collinear points, not an arbitrary set. For {{nowrap|1=''n'' = 2}} the space is the projective line, so all points are collinear and this is no restriction.
* {{nowrap|PGL(2, ''K'')}} does not act 4-transitively on the projective line (except for {{nowrap|PGL(2, 3)}}, as '''P'''<sup>1</sup>(3) has {{nowrap|1=3 + 1 = 4}} points, so 3-transitive implies 4-transitive); the invariant that is preserved is the [[cross ratio]], and this determines where every other point is sent: specifying where 3 points are mapped determines the map. Thus in particular it is not the full collineation group of the projective line (except for '''F'''<sub>2</sub> and '''F'''<sub>3</sub>).
* {{nowrap|PSL(2, ''q'')}} and {{nowrap|PGL(2, ''q'')}} (for {{nowrap|''q'' > 2}}, and ''q'' odd for PSL) are two of the four families of [[Zassenhaus group]]s.
* {{nowrap|PGL(''n'', ''K'')}} is an [[algebraic group]] of dimension {{nowrap|''n''<sup>2</sup> − 1}} and an open subgroup of the projective space '''P'''<sup>''n''<sup>2</sup>−1</sup>. As defined, the functor {{nowrap|PSL(''n'', ''K'')}} does not define an algebraic group, or even an fppf sheaf, and its sheafification in the [[fppf topology]] is in fact {{nowrap|PGL(''n'', ''K'')}}.
* PSL and PGL are [[centerless]] – this is because the diagonal matrices are not only the center, but also the [[hypercenter]] (the quotient of a group by its center is not necessarily centerless).<ref group="note">For PSL (except {{nowrap|PSL(2, 2)}} and {{nowrap|PSL(2, 3)}}) this follows by [[Grün's lemma]] because SL is a [[perfect group]] (hence center equals hypercenter), but for PGL and the two exceptional PSLs this requires additional checking.</ref>

=== Fractional linear transformations ===
{{details|Möbius transformation#Projective matrix representations}}
As for [[Möbius transformation]]s, the group {{nowrap|PGL(2, ''K'')}} can be interpreted as [[fractional linear transformation]]s with coefficients in ''K''. Points in the projective line over ''K'' correspond to pairs from ''K''<sup>2</sup>, with two pairs being equivalent when they are proportional. When the second coordinate is non-zero, a point can be represented by {{nowrap|[''z'', 1]}}. Then when {{nowrap|''ad'' − ''bc'' ≠ 0}}, the action of {{nowrap|PGL(2, ''K'')}} is by linear transformation:
: <math>[z,\ 1]\begin{pmatrix} a & c \\ b & d \end{pmatrix} \ = \ [az + b,\ cz + d] \ = \ \left [\frac{a z + b}{c z + d},\ 1\right ].</math>
In this way successive transformations can be written as right multiplication by such matrices, and [[matrix multiplication]] can be used for the group product in {{nowrap|PGL(2, ''K'')}}.