projective special linear group

In mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector space ''V'' on the associated projective space P(''V''). Explicitly, the projective linear group is the quotient group
: PGL(''V'') = GL(''V'')/Z(''V'')
where GL(''V'') is the general linear group of ''V'' and Z(''V'') 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:
: PSL(''V'') = SL(''V'')/SZ(''V'')
where SL(''V'') is the special linear group over ''V'' and SZ(''V'') is the subgroup of scalar transformations with unit determinant. Here SZ is the center of SL, and is naturally identified with the group of ''n''th roots of unity in ''F'' (where ''n'' is the dimension of ''V'' and ''F'' is the base field).

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 ''n''th 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 (''x''₀ : ''x''₁ : ... : ''x_n'') is the underlying group of the geometry.This is therefore for projective space of dimension ''n'' Stated differently, the natural action of GL(''V'') on ''V'' descends to an action of PGL(''V'') on the projective space ''P''(''V'').

The projective linear groups therefore generalise the case of Möbius transformations (sometimes called the Möbius group), which acts on the projective line.

Note that unlike the general linear group, which is generally defined axiomatically as "invertible functions preserving the linear (vector space) 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 (or more generally one-to-one) map which sends collinear points to collinear points. One can define a projective space axiomatically in terms of an incidence structure (a set of points ''P'', lines ''L'', and an incidence relation ''I'' specifying which points lie on which lines) 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,"Preserving the incidence relation" means that if point ''p'' is on line ''l'' then ''f''(''p'') is in ''g''(''l''); formally, if then . which is exactly a collineation of a space to itself. Projective linear transforms are collineations (planes in a vector space correspond to lines in the associated projective space, and linear transforms map planes to planes, so projective linear transforms map lines to lines), but in general not all collineations are projective linear transforms – PGL is in general a proper subgroup of the collineation group.

Specifically, for (a projective line), 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 (where PGL is the full symmetric group), 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 (F_''p'' or Q), we have , but for ''K'' a field with non-trivial Galois automorphisms (such as F_''p''^''n'' for or C), 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 (base point) 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(''K''/''k'') (for ).

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 (the elements of ), which corresponds to the stereographic projection of rotations of the unit hypersphere, and has dimension . Visually, this corresponds to standing at the origin (or placing a camera 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 (an element of SO(''n''), which has dimension .), 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 (it preserves projective lines), but it is not the full collineation group, which is instead either PΓL (for ) or the full symmetric group for (the projective line).
* Every ( biregular) 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 , 1 , 1 , 0 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 (except for , as P¹(3) has 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 F₂ and F₃).
* and (for , and ''q'' odd for PSL) 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 (the quotient of a group by its center is not necessarily centerless).For PSL (except and ) 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.

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 ''K''², 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:
: ,\ 1begin a & c \\ b & d \end \ = \ z + b,\ cz + d\ = \ \left frac,\ 1\right
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''(''q''). They are finite simple groups whenever ''n'' is at least 2, with two exceptions: ''L''₂(2), which is isomorphic to S₃, the symmetric group on 3 letters, and is solvable; and ''L''₂(3), 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 (unless and or 3).

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 (general ''n'', general finite field) 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
: (''q''^''n'' − 1)(''qⁿ'' − ''q'')(''qⁿ'' − ''q''²) â‹…â‹…â‹… (''q''^''n'' − ''q''^{''n''−1})/(gcd(''n'', ''q'' − 1)) = ''q''^{''n''2−1} − O(''q''^{''n''2−3}),
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 ''n''th root; and it is also the number of ''n''th roots of unity in F_''q''.These are equal because they are the kernel and cokernel of the endomorphism ; formally, . More abstractly, the first realizes PSL as SL/SZ, while the second realizes PSL as the kernel of .

Exceptional isomorphisms

In addition to the isomorphisms
: , , and ,
there are other exceptional isomorphisms between projective special linear groups and alternating groups (these groups are all simple, as the alternating group over 5 or more letters is simple):
: ''L''₂(4) ≅ A₅
: ''L''₂(5) ≅ A₅ (see ' for a proof)
: ''L''₂(9) ≅ A₆
: ''L''₄(2) ≅ A₈
The isomorphism allows one to see the exotic outer automorphism of A₆ in terms of field automorphism and matrix operations. The isomorphism is of interest in the structure of the Mathieu group M₂₄.

The associated extensions are covering groups of the alternating groups ( universal perfect central extensions) for A₄, A₅, by uniqueness of the universal perfect central extension; for , 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:
: , the alternating group on five elements, or equivalently the icosahedral group;
: , the symmetric group on five elements;
: the double cover of the alternating group A₅, or equivalently the binary icosahedral group.
They can also be used to give a construction of an exotic map , as described below. Note however that is not a double cover of S₅, but is rather a 4-fold cover.

A further isomorphism is:
: is the simple group of order 168, the second-smallest non-abelian simple group, and is not an alternating group; see '' PSL(2, 7)''.

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(4, 2) ≃ PSp(4, 3), 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: acts on the projective space P^{''n''−1}(''q''), which has points, and this yields a map from the projective linear group to the symmetric group on points. For , this is the projective line P¹(''q'') which has points, so there is a map .

To understand these maps, it is useful to recall these facts:
* The order of is
*: (''q''² − 1)(''q''² − ''q'')/(''q'' − 1) = ''q''³ − ''q'' = (''q'' − 1)''q''(''q'' + 1);
: the order of 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 and 3- 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:
* , of order 6, which is an isomorphism.
** The inverse map (a projective representation of S₃) can be realized by the anharmonic group, and more generally yields an embedding for all fields.
* , of orders 12 and 24, the latter of which is an isomorphism, with being the alternating group.
** The anharmonic group gives a partial map in the opposite direction, mapping as the stabilizer of the point −1.
* , of order 60, yielding the alternating group A₅.
* , of orders 60 and 120, which yields an embedding of S₅ (respectively, A₅) as a ''transitive'' subgroup of S₆ (respectively, A₆). This is an example of an exotic map , and can be used to construct the exceptional outer automorphism of S₆. Note that the isomorphism is not transparent from this presentation: there is no particularly natural set of 5 elements on which acts.

Action on ''p'' points

While naturally acts on points, non-trivial actions on fewer points are rarer. Indeed, for acts non-trivially on ''p'' points if and only if , 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.Since ''p'' divides the order of the group, the group does not embed in (or, since simple, map non-trivially to) ''S_k'' for , as ''p'' does not divide the order of this latter group. 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 (it is a non-trivial quotient, and the PSL group is not simple), while for 5, 7, and 11 the action is faithful (as the group is simple and the action is non-trivial), and yields an embedding into S_''p''. In all but the last case, , it corresponds to an exceptional isomorphism, where the right-most group has an obvious action on ''p'' points:
* ''L''₂(2) ≅ S₃ $\twoheadrightarrow$ S₂ via the sign map;
* ''L''₂(3) ≅ A₄ $\twoheadrightarrow$ A₃ ≅ C₃ via the quotient by the Klein 4-group;
* ''L''₂(5) ≅ A₅. To construct such an isomorphism, one needs to consider the group ''L''₂(5) as a Galois group of a Galois cover ''a''₅: , where ''X''(''N'') is a modular curve of level ''N''. This cover is ramified at 12 points. The modular curve X(5) has genus 0 and is isomorphic to a sphere over the field of complex numbers, and then the action of ''L''₂(5) on these 12 points becomes the symmetry group of an icosahedron. One then needs to consider the action of the symmetry group of icosahedron on the five associated tetrahedra.
* which acts on the points of the Fano plane (projective plane over F₂); this can also be seen as the action on order 2 biplane, which is the ''complementary'' Fano plane.
* ''L''₂(11) is subtler, and elaborated below; it acts on the order 3 biplane.
Further, ''L''₂(7) and ''L''₂(11) 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 have different point stabilizers); 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 (as sets, not as groups) of the groups as , , and , where the groups A₄, S₄ and A₅ are the isometry groups of the Platonic solids, and correspond to ''E''₆, ''E''₇, and ''E''₈ under the McKay correspondence. These three exceptional cases are also realized as the geometries of polyhedra (equivalently, tilings of Riemann surfaces), respectively: the compound of five tetrahedra inside the icosahedron (sphere, genus 0), the order 2 biplane (complementary Fano plane) inside the Klein quartic (genus 3), and the order 3 biplane ( Paley biplane) inside the buckyball surface (genus 70).

The action of ''L''₂(11) can be seen algebraically as due to an exceptional inclusion – there are two conjugacy classes of subgroups of ''L''₂(11) that are isomorphic to ''L''₂(5), each with 11 elements: the action of ''L''₂(11) by conjugation on these is an action on 11 points, and, further, the two conjugacy classes are related by an outer automorphism of ''L''₂(11). (The same is true for subgroups of ''L''₂(7) isomorphic to S₄, and this also has a biplane geometry.)

Geometrically, this action can be understood via a '' biplane geometry'', which is defined as follows. A biplane geometry is a symmetric design (a set of points and an equal number of "lines", or rather blocks) 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 (and two lines determining one point), they determine two lines (respectively, points). In this case (the Paley biplane, obtained from the Paley digraph of order 11), the points are the affine line (the finite field) F₁₁, where the first line is defined to be the five non-zero quadratic residues (points which are squares: 1, 3, 4, 5, 9), and the other lines are the affine translates of this (add a constant to all the points). ''L''₂(11) is then isomorphic to the subgroup of S₁₁ that preserve this geometry (sends lines to lines), 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 ''L''₂(5) is the stabilizer of a given line, or dually of a given point.

More surprisingly, the coset space ''L''₂(11)/(Z/11Z), which has order (and on which the icosahedral group acts) naturally has the structure of a buckeyball, which is used in the construction of the buckyball surface.

Mathieu groups

The group can be used to construct the Mathieu group M₂₄, one of the sporadic simple groups; in this context, one refers to 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 – meaning that it has 21 points, each line ("block", in Steiner terminology) has 5 points, and any 2 points determine a line – and on which acts. One calls this Steiner system W₂₁ ("W" for Witt), and then expands it to a larger Steiner system W₂₄, expanding the symmetry group along the way: to the projective general linear group , then to the projective semilinear group , and finally to the Mathieu group M₂₄.

M₂₄ also contains copies of , which is maximal in M₂₂, and , which is maximal in M₂₄, and can be used to construct M₂₄.Conway, Sloane, SPLAG

Hurwitz surfaces

PSL groups arise as Hurwitz groups (automorphism groups of Hurwitz surfaces – algebraic curves of maximal possibly symmetry group). The Hurwitz surface of lowest genus, the Klein quartic (genus 3), has automorphism group isomorphic to (equivalently ), while the Hurwitz surface of second-lowest genus, the Macbeath surface (genus 7), has automorphism group isomorphic to .

In fact, many but not all simple groups arise as Hurwitz groups (including the monster group, though not all alternating groups or sporadic groups), though PSL is notable for including the smallest such groups.

Modular group

The groups arise in studying the modular group, , 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 (and of the projective special linear group ) is the symmetries of the set In projective coordinates, the points are given by :1 :1 and :0 which explains why their stabilizer is represented by integral matrices. which is known as the anharmonic group, and arises as the symmetries of the six cross-ratios. The subgroup can be expressed as fractional linear transformations, or represented (non-uniquely) by matrices, as:
:
Note that the top row is the identity and the two 3-cycles, and are orientation-preserving, forming a subgroup in , while the bottom row is the three 2-cycles, and are in and , but not in , 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 under reduction mod ''n''. Notably, for , this subgroup maps isomorphically to ,This isomorphism can be seen by removing the minus signs in matrices, which yields the matrices for and thus provides a splitting for the quotient map .

The fixed points of both 3-cycles are the "most symmetric" cross-ratios, $e^ = \tfrac \pm \tfraci$, the solutions to (the primitive sixth roots of unity). The 2-cycles interchange these, as they do any points other than their fixed points, which realizes the quotient map by the group action on these two points. That is, the subgroup consisting of the identity and the 3-cycles, , fixes these two points, while the other elements interchange them.

The fixed points of the individual 2-cycles are, respectively, −1, 1/2, 2, and this set is also preserved and permuted by the 3-cycles. This corresponds to the action of S₃ on the 2-cycles (its Sylow 2-subgroups) by conjugation and realizes the isomorphism with the group of inner automorphisms, .

Geometrically, this can be visualized as the rotation group of the triangular bipyramid, which is isomorphic to the dihedral group of the triangle ; see anharmonic group.

Topology

Over the real and complex numbers, the topology of PGL and PSL can be determined from the fiber bundles that define them:
: $\begin \mathrm &\cong& K^\times &\to& \mathrm &\to& \mathrm \\ \mathrm &\cong& \mu_n &\to& \mathrm &\to& \mathrm \end$
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 ''n''th 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 , so up to homotopy, is a 2-fold covering space, and all higher homotopy groups agree.

For PGL over the complexes, the fiber is , so up to homotopy, is a circle bundle. The higher homotopy groups of the circle vanish, so the homotopy groups of and agree for . In fact, ''Ï€''₂ always vanishes for Lie groups, so the homotopy groups agree for . For , we have that . The fundamental group of is a finite cyclic group of order 2.

Covering groups

Over the real and complex numbers, the projective special linear groups are the ''minimal'' ( centerless) Lie group realizations for the special linear Lie algebra $\mathfrak(n)\colon$ every connected Lie group whose Lie algebra is $\mathfrak(n)$ is a cover of . Conversely, its universal covering group is the ''maximal'' (simply connected) element, and the intermediary realizations form a lattice of covering groups.

For example, has center and fundamental group Z, and thus has universal cover and covers the centerless .

Representation theory

A group homomorphism from a group ''G'' to a projective linear group is called a projective representation of the group ''G'', by analogy with a linear representation (a homomorphism ). 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 , though it can be defined for low dimensions.

For (or in fact ) the projective space of ''K''⁰ is empty, as there are no 1-dimensional subspaces of a 0-dimensional space. Thus, 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 is trivial, rather than an inclusion as it is in higher dimensions.

For , the projective space of ''K''¹ is a single point, as there is a single 1-dimensional subspace. Thus, 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 being trivial.

For , is non-trivial, but is unusual in that it is 3-transitive, unlike higher dimensions when it is only 2-transitive.

Examples

* PSL(2, 7)
* Modular group,
* PSL(2, R)
* Möbius group,

Subgroups

* Projective orthogonal group, PO – maximal compact subgroup of PGL
* Projective unitary group, PU
* Projective special orthogonal group, PSO – maximal compact subgroup of PSL
* Projective special unitary group, PSU

Larger groups

The projective linear group is contained within larger groups, notably:
* Projective semilinear group, PΓL, which allows field automorphisms.
* Cremona group, ''Cr''(P^''n''(''k'')) of birational automorphisms; any biregular automorphism is linear, so PGL coincides with the group of biregular automorphisms.

See also

* Projective transformation
* Unit

Notes

References

*

{{DEFAULTSORT:Projective Linear Group
Lie groups
Projective geometry