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 PGL 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: PGL is the group associated to GL, 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 n = 2, 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 n ≥ 3, the collineation group is the projective semilinear group, PΓL – this is PGL, twisted by field automorphisms; formally, PΓL ≅ PGL ⋊ Gal, where k is the prime field for K; this is the fundamental theorem of projective geometry. Thus for K a prime field, we have PGL = PΓL, 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 PΓL/PGL = Gal 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

As for Möbius transformations, the group PGL 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 ad– bc ≠ 0, the action of PGL 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 PGL.

Finite fields

The projective special linear groups PSL for a finite field Fq are often written as PSL or Ln. 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 SL are thus quasisimple: perfect central extensions of a simple group.

History

The groups PSL 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, GL, in studying the Galois group of the general equation of degree pν.
The groups PSL were then constructed in the classic 1870 text by Camille Jordan, Traité des substitutions et des équations algébriques.

Order

The order of PGL 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 above, divided by, the number of scalar matrices with determinant 1 – or equivalently dividing by, the number of classes of element that have no nth root, or equivalently, dividing by the number of nth roots of unity in Fq.

Exceptional isomorphisms

In addition to the isomorphisms
there are other exceptional isomorphisms between projective special linear groups and alternating groups :
The isomorphism L2 ≅ A6 allows one to see the exotic outer automorphism of A6 in terms of field automorphism and matrix operations. The isomorphism L4 ≅ A8 is of interest in the structure of the Mathieu group M24.
The associated extensions SL → PSL are covering groups of the alternating groups for A4, A5, by uniqueness of the universal perfect central extension; for L2 ≅ A6, the associated extension is a perfect central extension, but not universal: there is a 3-fold covering group.
The groups over F5 have a number of exceptional isomorphisms:
They can also be used to give a construction of an exotic map S5 → S6, as described below. Note however that GL is not a double cover of S5, but is rather a 4-fold cover.
A further isomorphism is:
The above exceptional isomorphisms involving the projective special linear groups are almost all of the exceptional isomorphisms between families of finite simple groups; the only other exceptional isomorphism is PSU ≃ PSp, between a projective special unitary group and a projective symplectic group.

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: PGL acts on the projective space Pn−1, which has / points, and this yields a map from the projective linear group to the symmetric group on / points. For n = 2, this is the projective line P1 which has / = q+1 points, so there is a map PGL → Sq+1.
To understand these maps, it is useful to recall these facts:
Thus the image is a 3-transitive subgroup of known order, which allows it to be identified. This yields the following maps:
While PSL naturally acts on / = 1+q+...+qn−1 points, non-trivial actions on fewer points are rarer. Indeed, for PSL acts non-trivially on p points if and only if p = 2, 3, 5, 7, or 11; for 2 and 3 the group is not simple, while for 5, 7, and 11, the group is simple – further, it does not act non-trivially on fewer than p points. This was first observed by Évariste Galois in his last letter to Chevalier, 1832.
This can be analyzed as follows; note that for 2 and 3 the action is not faithful, while for 5, 7, and 11 the action is faithful, and yields an embedding into Sp. In all but the last case, PSL, it corresponds to an exceptional isomorphism, where the right-most group has an obvious action on p points:
Further, L2 and L2 have two inequivalent actions on p points; geometrically this is realized by the action on a biplane, which has p points and p blocks – the action on the points and the action on the blocks are both actions on p points, but not conjugate ; they are instead related by an outer automorphism of the group.
More recently, these last three exceptional actions have been interpreted as an example of the ADE classification: these actions correspond to products of the groups as A4 × Z/5Z, S4 × Z/7Z, and A5 × Z/11Z, where the groups A4, S4 and A5 are the isometry groups of the Platonic solids, and correspond to E6, E7, and E8 under the McKay correspondence. These three exceptional cases are also realized as the geometries of polyhedra, respectively: the compound of five tetrahedra inside the icosahedron, the order 2 biplane inside the Klein quartic, and the order 3 biplane inside the buckyball surface.
The action of L2 can be seen algebraically as due to an exceptional inclusion – there are two conjugacy classes of subgroups of L2 that are isomorphic to L2, each with 11 elements: the action of L2 by conjugation on these is an action on 11 points, and, further, the two conjugacy classes are related by an outer automorphism of L2.
Geometrically, this action can be understood via a biplane geometry, which is defined as follows. A biplane geometry is a symmetric design such that any set of two points is contained in two lines, while any two lines intersect in two points; this is similar to a finite projective plane, except that rather than two points determining one line, they determine two lines. In this case, the points are the affine line F11, where the first line is defined to be the five non-zero quadratic residues, and the other lines are the affine translates of this. L2 is then isomorphic to the subgroup of S11 that preserve this geometry, giving a set of 11 points on which it acts – in fact two: the points or the lines, which corresponds to the outer automorphism – while L2 is the stabilizer of a given line, or dually of a given point.
More surprisingly, the coset space L2/Z/11Z, which has order 660/11 = 60 naturally has the structure of a buckeyball, which is used in the construction of the buckyball surface.

Mathieu groups

The group PSL can be used to construct the Mathieu group M24, one of the sporadic simple groups; in this context, one refers to PSL as M21, though it is not properly a Mathieu group itself. One begins with the projective plane over the field with four elements, which is a Steiner system of type S – meaning that it has 21 points, each line has 5 points, and any 2 points determine a line – and on which PSL acts. One calls this Steiner system W21, and then expands it to a larger Steiner system W24, expanding the symmetry group along the way: to the projective general linear group PGL, then to the projective semilinear group PΓL, and finally to the Mathieu group M24.
M24 also contains copies of PSL, which is maximal in M22, and PSL, which is maximal in M24, and can be used to construct M24.

Hurwitz surfaces

PSL groups arise as Hurwitz groups. The Hurwitz surface of lowest genus, the Klein quartic, has automorphism group isomorphic to PSL, while the Hurwitz surface of second-lowest genus, the Macbeath surface, has automorphism group isomorphic to PSL.
In fact, many but not all simple groups arise as Hurwitz groups, though PSL is notable for including the smallest such groups.

Modular group

The groups PSL arise in studying the modular group, PSL, as quotients by reducing all elements mod n; the kernels are called the principal congruence subgroups.
A noteworthy subgroup of the projective general linear group PGL is the symmetries of the set ⊂ P1 these also occur in the six cross-ratios. The subgroup can be expressed as fractional linear transformations, or represented by matrices, as:
Note that the top row is the identity and the two 3-cycles, and are orientation-preserving, forming a subgroup in PSL, while the bottom row is the three 2-cycles, and are in PGL and PSL, but not in PSL, hence realized either as matrices with determinant −1 and integer coefficients, or as matrices with determinant 1 and Gaussian integer coefficients.
This maps to the symmetries of ⊂ P1 under reduction mod n. Notably, for n = 2, this subgroup maps isomorphically to PGL = PSL ≅ S3, and thus provides a splitting for the quotient map
A further property of this subgroup is that the quotient map S3 → S2 is realized by the group action. That is, the subgroup C3 < S3 consisting of the 3-cycles and the identity stabilizes the golden ratio and inverse golden ratio while the 2-cycles interchange these, thus realizing the map.
The fixed points of the individual 2-cycles are, respectively, −1, 1/2, 2, and this set is also preserved and permuted, corresponding to the action of S3 on the 2-cycles by conjugation and realizing the isomorphism

Topology

Over the real and complex numbers, the topology of PGL and PSL can be determined from the fiber bundles that define them:
via the long exact sequence of a fibration.
For both the reals and complexes, SL is a covering space of PSL, with number of sheets equal to the number of nth roots in K; thus in particular all their higher homotopy groups agree. For the reals, SL is a 2-fold cover of PSL for n even, and is a 1-fold cover for n odd, i.e., an isomorphism:
For the complexes, SL is an n-fold cover of PSL.
For PGL, for the reals, the fiber is R* ≅, so up to homotopy, GL → PGL is a 2-fold covering space, and all higher homotopy groups agree.
For PGL over the complexes, the fiber is C* ≅ S1, so up to homotopy, GL → PGL is a circle bundle. The higher homotopy groups of the circle vanish, so the homotopy groups of GL and PGL agree for n ≥ 3. In fact, π2 always vanishes for Lie groups, so the homotopy groups agree for n ≥ 2. For n = 1, we have that π1 = π1 = Z and hence PGL is simply connected.

Covering groups

Over the real and complex numbers, the projective special linear groups are the minimal Lie group realizations for the special linear Lie algebra every connected Lie group whose Lie algebra is is a cover of PSL. Conversely, its universal covering group is the maximal element, and the intermediary realizations form a lattice of covering groups.
For example, SL has center and fundamental group Z, and thus has universal cover and covers the centerless PSL.

Representation theory

A group homomorphism G → PGL from a group G to a projective linear group is called a projective representation of the group G, by analogy with a linear representation. These were studied by Issai Schur, who showed that projective representations of G can be classified in terms of linear representations of central extensions of G. This led to the Schur multiplier, which is used to address this question.

Low dimensions

The projective linear group is mostly studied for n ≥ 2, though it can be defined for low dimensions.
For n = 0 the projective space of K0 is empty, as there are no 1-dimensional subspaces of a 0-dimensional space. Thus, PGL 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 K* → GL is trivial, rather than an inclusion as it is in higher dimensions.
For n = 1, the projective space of K1 is a single point, as there is a single 1-dimensional subspace. Thus, PGL 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 is an isomorphism, corresponding to PGL := GL/K* ≅ being trivial.
For n = 2, PGL is non-trivial, but is unusual in that it is 3-transitive, unlike higher dimensions when it is only 2-transitive.

Examples

The projective linear group is contained within larger groups, notably: