Projective linear group

In mathematics, especially in the group theoretic area of algebra, the projective linear group is the induced action of the general linear group of a vector space V on the associated projective space P. Explicitly, the projective linear group is the quotient group
where GL is the general linear group of V and Z is the subgroup of all nonzero scalar transformations of V; these are quotiented out because they act trivially on the projective space and they form the kernel of the action, and the notation "Z" reflects that the scalar transformations form the 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:
where SL is the special linear group over V and SZ is the subgroup of scalar transformations with unit determinant. Here SZ is the center of SL, and is naturally identified with the group of nth roots of unity in F.
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, the alternate notations and are also used.
Note that and are isomorphic if and only if every element of F has an nth root in F. As an example, note that, but that ; 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, with an important example being the modular group,.

Name

The name comes from projective geometry, where the projective group acting on homogeneous coordinates is the underlying group of the geometry. Stated differently, the natural action of GL on V descends to an action of PGL on the projective space P.
The projective linear groups therefore generalise the case of Möbius transformations, which acts on the projective line.
Note that unlike the general linear group, which is generally defined axiomatically as "invertible functions preserving the linear 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: is the group associated to, and is the projective linear group of -dimensional projective space, not n-dimensional projective space.

Collineations

A related group is the collineation group, which is defined axiomatically. A collineation is an invertible map which sends collinear points to collinear points. One can define a projective space axiomatically in terms of an incidence structure 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, which is exactly a collineation of a space to itself. Projective linear transforms are collineations, but in general not all collineations are projective linear transforms – PGL is in general a proper subgroup of the collineation group.
Specifically, for , all points are collinear, so the collineation group is exactly the symmetric group of the points of the projective line, and except for F₂ and F₃, PGL is a proper subgroup of the full symmetric group on these points.
For, the collineation group is the projective semilinear group, PΓL – this is PGL, twisted by field automorphisms; formally,, where k is the prime field for K; this is the fundamental theorem of projective geometry. Thus for K a prime field, we have, but for K a field with non-trivial Galois automorphisms, 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 corresponds to "choices of linear structure", with the identity 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 planes, 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.

Elements

The elements of the projective linear group can be understood as "tilting the plane" along one of the axes, and then projecting to the original plane, and also have dimension n.
A more familiar geometric way to understand the projective transforms is via projective rotations, which corresponds to the stereographic projection of rotations of the unit hypersphere, and has dimension. Visually, this corresponds to standing at the origin, and turning one's angle of view, then projecting onto a flat plane. Rotations in axes perpendicular to the hyperplane preserve the hyperplane and yield a rotation of the hyperplane, while rotations in axes parallel to the hyperplane are proper projective maps, and accounts for the remaining n dimensions.

Properties

PGL sends collinear points to collinear points, but it is not the full collineation group, which is instead either PΓL or the full symmetric group for .
Every algebraic automorphism of a projective space is projective linear. The birational automorphisms form a larger group, the Cremona group.
PGL acts faithfully on projective space: non-identity elements act non-trivially. 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-transitively on projective space. 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.
acts sharply 3-transitively on the projective line. Three arbitrary points are conventionally mapped to , , ; in alternative notation, 0, 1, ∞. In fractional linear transformation notation, the function maps,,, and is the unique such map that does so. This is the cross-ratio – see ' for details.
For, does not act 3-transitively, because it must send 3 collinear points to 3 other collinear points, not an arbitrary set. For the space is the projective line, so all points are collinear and this is no restriction.
does not act 4-transitively on the projective line ; 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.
and are two of the four families of Zassenhaus groups.
is an algebraic group of dimension and an open subgroup of the projective space P'n''²−1. As defined, the functor does not define an algebraic group, or even an fppf sheaf, and its sheafification in the fppf topology is in fact.
PSL and PGL are centerless – this is because the diagonal matrices are not only the center, but also the hypercenter.
Fractional linear transformations

As for Möbius transformations, the group can be interpreted as fractional linear transformations with coefficients in K. Points in the projective line over K correspond to pairs from K², with two pairs being equivalent when they are proportional. When the second coordinate is non-zero, a point can be represented by. Then when, the action of is by linear transformation:
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.

Finite fields

The projective special linear groups for a finite field F_q are often written as or L_n. They are finite simple groups whenever n is at least 2, with two exceptions: L₂, which is isomorphic to S₃, the symmetric group on 3 letters, and is solvable; and L₂, which is isomorphic to A₄, the alternating group on 4 letters, and is also solvable. These exceptional isomorphisms can be understood as arising from the [|action on the projective line].
The special linear groups are thus quasisimple: perfect central extensions of a simple group.

History

The groups for any prime number p were constructed by Évariste Galois in the 1830s, and were the second family of finite simple groups, after the alternating groups. Galois constructed them as fractional linear transforms, and observed that they were simple except if p was 2 or 3; this is contained in his last letter to Chevalier. In the same letter and attached manuscripts, Galois also constructed the general linear group over a prime field,, in studying the Galois group of the general equation of degree p^ν.
The groups for any prime power q were then constructed in the classic 1870 text by Camille Jordan, ''Traité des substitutions et des équations algébriques.''

Order

The order of is
which corresponds to the order of, divided by for projectivization; see q-analog for discussion of such formulas. Note that the degree is, which agrees with the dimension as an algebraic group. The "O" is for big O notation, meaning "terms involving lower order". This also equals the order of ; there dividing by is due to the determinant.
The order of is the order of as above, divided by. This is equal to, the number of scalar matrices with determinant 1;, the number of classes of element that have no nth root; and it is also the number of nth roots of unity in F_q.