Inmathematics, especially in thegroup theoretic area ofalgebra, theprojective linear group (also known as theprojective general linear group or PGL) is the inducedaction of thegeneral linear group of avector spaceV on the associatedprojective space P(V). Explicitly, the projective linear group is thequotient group

PGL(V) = GL(V) / Z(V)

where GL(V) is thegeneral linear group ofV and Z(V) is the subgroup of all nonzeroscalar transformations ofV; these are quotiented out because they acttrivially on the projective space and they form thekernel of the action, and the notation "Z" reflects that the scalar transformations form thecenter of the general linear group.

Theprojective special linear group, PSL, is defined analogously, as the induced action of thespecial linear group on the associated projective space. Explicitly:

PSL(V) = SL(V) / SZ(V)

where SL(V) is the special linear group overV and SZ(V) is the subgroup of scalar transformations with unitdeterminant. Here SZ is the center of SL, and is naturally identified with the group ofnthroots of unity inF (wheren is thedimension ofV andF is the basefield).

PGL and PSL are some of the fundamental groups of study, part of the so-calledclassical groups, and an element of PGL is calledprojective linear transformation,projective transformation orhomography. IfV is then-dimensional vector space over a fieldF, namelyV =Fⁿ, the alternate notationsPGL(n,F) andPSL(n,F) are also used.

Note thatPGL(n,F) andPSL(n,F) areisomorphicif and only if every element ofF has annth root in F. As an example, note thatPGL(2,C) = PSL(2,C), but thatPGL(2,R) > PSL(2,R);^[1] 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 aring, with an important example being themodular group,PSL(2,Z).

The name comes fromprojective geometry, where the projective group acting onhomogeneous coordinates (x₀ :x₁ : ... :x_n) is the underlying group of the geometry.^{[note 1]} Stated differently, the naturalaction of GL(V) onV descends to an action of PGL(V) on the projective spaceP(V).

The projective linear groups therefore generalise the casePGL(2,C) ofMöbius transformations (sometimes called theMöbius group), which acts on theprojective 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 definedconstructively, 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(n,F) is the group associated toGL(n,F), and is the projective linear group of(n − 1)-dimensional projective space, notn-dimensional projective space.

A related group is thecollineation group, which is defined axiomatically. A collineation is an invertible (or more generally one-to-one) map which sendscollinear points to collinear points. One candefine a projective space axiomatically in terms of anincidence structure (a set of pointsP, linesL, and anincidence relationI specifying which points lie on which lines) satisfying certain axioms – an automorphism of a projective space thus defined then being an automorphismf of the set of points and an automorphismg of the set of lines, preserving the incidence relation,^{[note 2]} 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, forn = 2 (a projective line), all points are collinear, so the collineation group is exactly thesymmetric group of the points of the projective line, and except forF₂ andF₃ (where PGL is the full symmetric group), PGL is a proper subgroup of the full symmetric group on these points.

Forn ≥ 3, the collineation group is theprojective semilinear group, PΓL – this is PGL, twisted byfield automorphisms; formally,PΓL ≅ PGL ⋊ Gal(K / k), wherek is theprime field forK; this is thefundamental theorem of projective geometry. Thus forK a prime field (F_p orQ), we havePGL = PΓL, but forK a field with non-trivial Galois automorphisms (such asF_pⁿ forn ≥ 2 orC), the projective linear group is a proper subgroup of the collineation group, which can be thought of as "transforms preserving a projectivesemi-linear structure". Correspondingly, the quotient groupPΓL / PGL = Gal(K / k) 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 projectivelinear transform. However, with the exception of thenon-Desarguesian planes, all projective spaces are the projectivization of a linear space over adivision ring though, as noted above, there are multiple choices of linear structure, namely atorsor over Gal(K / k) (forn ≥ 3).

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 dimensionn.

A more familiar geometric way to understand the projective transforms is viaprojective rotations (the elements ofPSO(n + 1)), which corresponds to thestereographic projection of rotations of the unit hypersphere, and has dimension⁠${\displaystyle 1+2+\cdots +n=(\genfrac{}{}{0ex}{}{n+1}{2})}$⁠. 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 thehyperplane preserve the hyperplane and yield a rotation of the hyperplane (an element of SO(n), which has dimension⁠${\displaystyle 1+2+\cdots +(n-1)=(\genfrac{}{}{0ex}{}{n}{2})}$⁠.), while rotations in axes parallel to the hyperplane are proper projective maps, and accounts for the remainingn dimensions.

• PGL sends collinear points to collinear points (it preserves projective lines), but it is not the fullcollineation group, which is instead either PΓL (forn > 2) or the fullsymmetric group forn = 2 (the projective line).
• Every (biregular) algebraic automorphism of a projective space is projective linear. Thebirational automorphisms form a larger group, theCremona 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 acts2-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 arelinearly independent, and GL acts transitively onk-element sets of linearly independent vectors.
• PGL(2,K) acts sharply 3-transitively on the projective line.
  Three arbitrary points are conventionally mapped to [0, 1], [1, 1], [1, 0]; in alternative notation, 0, 1, ∞. In fractional linear transformation notation, the function⁠x −a/x −c⁠ ⋅⁠b −c/b −a⁠ mapsa ↦ 0,b ↦ 1,c ↦ ∞, and is the unique such map that does so. This is thecross-ratio(x,b;a,c) – seeCross-ratio § Transformational approach for details.
• Forn ≥ 3,PGL(n,K) does not act 3-transitively, because it must send 3 collinear points to 3 other collinear points, not an arbitrary set. Forn = 2 the space is the projective line, so all points are collinear and this is no restriction.
• PGL(2,K) does not act 4-transitively on the projective line (except forPGL(2, 3), asP¹(3) has3 + 1 = 4 points, so 3-transitive implies 4-transitive); the invariant that is preserved is thecross 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 forF₂ andF₃).
• PSL(2,q) andPGL(2,q) (forq > 2, andq odd for PSL) are two of the four families ofZassenhaus groups.
• PGL(n,K) is analgebraic group of dimensionn² − 1 and an open subgroup of the projective spacePⁿ²⁻¹. As defined, the functorPSL(n,K) does not define an algebraic group, or even an fppf sheaf, and its sheafification in thefppf topology is in factPGL(n,K).
• PSL and PGL arecenterless – this is because the diagonal matrices are not only the center, but also thehypercenter (the quotient of a group by its center is not necessarily centerless).^{[note 3]}

As forMöbius transformations, the groupPGL(2,K) can be interpreted asfractional linear transformations with coefficients inK. Points in the projective line overK correspond to pairs fromK², with two pairs being equivalent when they are proportional. When the second coordinate is non-zero, a point can be represented by[z, 1]. Then whenad −bc ≠ 0, the action ofPGL(2,K) is by linear transformation:

In this way successive transformations can be written as right multiplication by such matrices, andmatrix multiplication can be used for the group product inPGL(2,K).

The projective special linear groupsPSL(n,F_q) for afinite fieldF_q are often written asPSL(n,q) orL_n(q). They arefinite simple groups whenevern is at least 2, with two exceptions:^[2]L₂(2), which is isomorphic to S₃, thesymmetric group on 3 letters, and issolvable; andL₂(3), which is isomorphic to A₄, thealternating group on 4 letters, and is also solvable. These exceptional isomorphisms can be understood as arising from theaction on the projective line.

The special linear groupsSL(n,q) are thusquasisimple: perfect central extensions of a simple group (unlessn = 2 andq = 2 or 3).

The groupsPSL(2,p) for any prime number p were constructed byÉvariste Galois in the 1830s, and were the second family of finitesimple groups, after thealternating groups.^[3] Galois constructed them as fractional linear transforms, and observed that they were simple except ifp was 2 or 3; this is contained in his last letter to Chevalier.^[4] In the same letter and attached manuscripts, Galois also constructed thegeneral linear group over a prime field,GL(ν,p), in studying the Galois group of the general equation of degreep^ν.

The groupsPSL(n,q) (generaln, general finite field) for any prime powerq were then constructed in the classic 1870 text byCamille Jordan,Traité des substitutions et des équations algébriques.

The order ofPGL(n,q) is

(qⁿ − 1)(qⁿ −q)(qⁿ −q²) ⋅⋅⋅ (qⁿ −qⁿ⁻¹)/(q − 1) =qⁿ²⁻¹ − O(qⁿ²⁻³),

which corresponds to theorder ofGL(n,q), divided byq − 1 for projectivization; seeq-analog for discussion of such formulas. Note that the degree isn² − 1, which agrees with the dimension as an algebraic group. The "O" is forbig O notation, meaning "terms involving lower order". This also equals the order ofSL(n,q); there dividing byq − 1 is due to the determinant.

The order ofPSL(n,q) is the order ofPGL(n,q) as above, divided bygcd(n,q − 1). This is equal to|SZ(n,q)|, the number of scalar matrices with determinant 1; |F^× / (F^×)ⁿ|, the number of classes of element that have nonth root; and it is also the number ofnthroots of unity inF_q.^{[note 4]}

In addition to the isomorphisms

L₂(2) ≅ S₃,L₂(3) ≅ A₄, andPGL(2, 3) ≅ S₄,

there are otherexceptional 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):

L₂(4) ≅ A₅

L₂(5) ≅ A₅ (see§ Action on p points for a proof)

L₂(9) ≅ A₆

L₄(2) ≅ A₈^[5]

The isomorphismL₂(9) ≅ A₆ allows one to see theexotic outer automorphism of A₆ in terms offield automorphism and matrix operations. The isomorphismL₄(2) ≅ A₈ is of interest in thestructure of the Mathieu group M₂₄.

The associated extensionsSL(n,q) → PSL(n,q) arecovering groups of the alternating groups (universal perfect central extensions) for A₄, A₅, by uniqueness of the universal perfect central extension; forL₂(9) ≅ A₆, the associated extension is a perfect central extension, but not universal: there is a 3-foldcovering group.

The groups overF₅ have a number of exceptional isomorphisms:

PSL(2, 5) ≅ A₅ ≅I, the alternating group on five elements, or equivalently theicosahedral group;

PGL(2, 5) ≅ S₅, thesymmetric group on five elements;

SL(2, 5) ≅ 2 ⋅ A₅ ≅ 2I thedouble cover of the alternating group A₅, or equivalently thebinary icosahedral group.

They can also be used to give a construction of anexotic mapS₅ → S₆, as described below. Note however thatGL(2, 5) is not a double cover of S₅, but is rather a 4-fold cover.

A further isomorphism is:

L₂(7) ≅L₃(2) is the simple group of order 168, the second-smallest non-abelian simple group, and is not an alternating group; seePSL(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 aprojective special unitary group and aprojective symplectic group.^[3]

Some of the above maps can be seen directly in terms of the action of PSL and PGL on the associated projective line:PGL(n,q) acts on the projective spacePⁿ⁻¹(q), which has(qⁿ − 1)/(q − 1) points, and this yields a map from the projective linear group to the symmetric group on(qⁿ − 1)/(q − 1) points. Forn = 2, this is the projective lineP¹(q) which has(q² − 1)/(q − 1) =q + 1 points, so there is a mapPGL(2,q) → S_q+1.

To understand these maps, it is useful to recall these facts:

• The order ofPGL(2,q) is

  (q² − 1)(q² −q)/(q − 1) =q³ −q = (q − 1)q(q + 1);

the order ofPSL(2,q) 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:

• PSL(2, 2) = PGL(2, 2) → S₃, of order 6, which is an isomorphism.
  • The inverse map (a projectiverepresentation of S₃) can be realized by theanharmonic group, and more generally yields an embeddingS₃ → PGL(2,q) for all fields.
• PSL(2, 3) < PGL(2, 3) → S₄, of orders 12 and 24, the latter of which is an isomorphism, withPSL(2, 3) being the alternating group.
  • The anharmonic group gives a partial map in the opposite direction, mappingS₃ → PGL(2, 3) as the stabilizer of the point −1.
• PSL(2, 4) = PGL(2, 4) → S₅, of order 60, yielding the alternating group A₅.
• PSL(2, 5) < PGL(2, 5) → S₆, of orders 60 and 120, which yields an embedding of S₅ (respectively, A₅) as atransitive subgroup of S₆ (respectively, A₆). This is an example of anexotic mapS₅ → S₆, and can be used to construct theexceptional outer automorphism of S₆.^[6] Note that the isomorphismPGL(2, 5) ≅ S₅ is not transparent from this presentation: there is no particularly natural set of 5 elements on whichPGL(2, 5) acts.

WhilePSL(n,q) naturally acts on(qⁿ − 1)/(q − 1) = 1 +q + ... +qⁿ⁻¹ points, non-trivial actions on fewer points are rarer. Indeed, forPSL(2,p) acts non-trivially onp points if and only ifp = 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 onfewer thanp points.^{[note 5]} This was first observed byÉvariste Galois in his last letter to Chevalier, 1832.^[7]

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,PSL(2, 11), it corresponds to an exceptional isomorphism, where the right-most group has an obvious action onp points:

• L₂(2) ≅ S₃${\displaystyle \twoheadrightarrow }$ S₂ via the sign map;
• L₂(3) ≅ A₄${\displaystyle \twoheadrightarrow }$ A₃ ≅ C₃ via the quotient by the Klein 4-group;
• L₂(5) ≅ A₅. To construct such an isomorphism, one needs to consider the groupL₂(5) as a Galois group of a Galois covera₅:X(5) →X(1) =P¹, whereX(N) is amodular curve of levelN. 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 ofL₂(5) on these 12 points becomes thesymmetry group of an icosahedron. One then needs to consider the action of the symmetry group of icosahedron on thefive associated tetrahedra.
• L₂(7) ≅L₃(2) which acts on the1 + 2 + 4 = 7 points of theFano plane (projective plane overF₂); this can also be seen as the action on order 2biplane, which is thecomplementary Fano plane.
• L₂(11) is subtler, and elaborated below; it acts on the order 3 biplane.^[8]

Further,L₂(7) andL₂(11) have twoinequivalent actions onp points; geometrically this is realized by the action on a biplane, which hasp points andp blocks – the action on the points and the action on the blocks are both actions onp points, but not conjugate (they have different point stabilizers); they are instead related by an outer automorphism of the group.^[9]

More recently, these last three exceptional actions have been interpreted as an example of theADE classification:^[10] these actions correspond to products (as sets, not as groups) of the groups asA₄ ×Z / 5Z,S₄ ×Z / 7Z, andA₅ ×Z / 11Z, where the groups A₄, S₄ and A₅ are the isometry groups of thePlatonic solids, and correspond toE₆,E₇, andE₈ under theMcKay correspondence. These three exceptional cases are also realized as the geometries of polyhedra (equivalently, tilings ofRiemann surfaces), respectively: thecompound of five tetrahedra inside the icosahedron (sphere, genus 0), the order 2 biplane (complementaryFano plane) inside the Klein quartic (genus 3), and the order 3 biplane (Paley biplane) inside thebuckyball surface (genus 70).^[11][12]

The action ofL₂(11) can be seen algebraically as due to an exceptional inclusionL₂(5)${\displaystyle \hookrightarrow }$L₂(11) – there are two conjugacy classes of subgroups ofL₂(11) that are isomorphic toL₂(5), each with 11 elements: the action ofL₂(11) by conjugation on these is an action on 11 points, and, further, the two conjugacy classes are related by an outer automorphism ofL₂(11). (The same is true for subgroups ofL₂(7) isomorphic to S₄, and this also has a biplane geometry.)

Geometrically, this action can be understood via abiplane geometry, which is defined as follows. A biplane geometry is asymmetric 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 (thePaley biplane, obtained from thePaley 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-zeroquadratic 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).L₂(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 – whileL₂(5) is the stabilizer of a given line, or dually of a given point.

More surprisingly, the coset spaceL₂(11) / (Z / 11Z), which has order660/11 = 60 (and on which the icosahedral group acts) naturally has the structure of abuckeyball, which is used in the construction of thebuckyball surface.

The groupPSL(3, 4) can be used to construct theMathieu group M₂₄, one of thesporadic simple groups; in this context, one refers toPSL(3, 4) 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 aSteiner system of typeS(2, 5, 21) – meaning that it has 21 points, each line ("block", in Steiner terminology) has 5 points, and any 2 points determine a line – and on whichPSL(3, 4) acts. One calls this Steiner system W₂₁ ("W" forWitt), and then expands it to a larger Steiner system W₂₄, expanding the symmetry group along the way: to the projective general linear groupPGL(3, 4), then to theprojective semilinear groupPΓL(3, 4), and finally to the Mathieu group M₂₄.

M₂₄ also contains copies ofPSL(2, 11), which is maximal in M₂₂, andPSL(2, 23), which is maximal in M₂₄, and can be used to construct M₂₄.^[13]

PSL groups arise asHurwitz groups (automorphism groups ofHurwitz surfaces – algebraic curves of maximal possibly symmetry group). The Hurwitz surface of lowest genus, theKlein quartic (genus 3), has automorphism group isomorphic toPSL(2, 7) (equivalentlyGL(3, 2)), while the Hurwitz surface of second-lowest genus, theMacbeath surface (genus 7), has automorphism group isomorphic toPSL(2, 8).

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

The groupsPSL(2,Z / nZ) arise in studying themodular group,PSL(2,Z), as quotients by reducing all elements modn; the kernels are called theprincipal congruence subgroups.

A noteworthy subgroup of the projectivegeneral linear groupPGL(2,Z) (and of the projective special linear groupPSL(2,Z[i])) is the symmetries of the set{0, 1, ∞} ⊂P¹(C)^{[note 6]} which is known as theanharmonic group, and arises as the symmetries of thesix cross-ratios. The subgroup can be expressed asfractional linear transformations, or represented (non-uniquely) by matrices, as:

${\displaystyle x}$	${\displaystyle 1/(1-x)}$	${\displaystyle (x-1)/x}$
${\displaystyle 1/x}$	${\displaystyle 1-x}$	${\displaystyle x/(x-1)}$

Note that the top row is the identity and the two 3-cycles, and are orientation-preserving, forming a subgroup inPSL(2,Z), while the bottom row is the three 2-cycles, and are inPGL(2,Z) andPSL(2,Z[i]), but not inPSL(2,Z), hence realized either as matrices with determinant −1 and integer coefficients, or as matrices with determinant 1 andGaussian integer coefficients.

This maps to the symmetries of{0, 1, ∞} ⊂P¹(n) under reduction modn. Notably, forn = 2, this subgroup maps isomorphically toPGL(2,Z / 2Z) = PSL(2,Z / 2Z) ≅ S₃,^{[note 7]} and thus provides a splittingPGL(2,Z / 2Z)${\displaystyle \hookrightarrow }$ PGL(2,Z) for the quotient mapPGL(2,Z)${\displaystyle \twoheadrightarrow }$ PGL(2,Z / 2Z).

The fixed points of both 3-cycles are the "most symmetric" cross-ratios,${\displaystyle e^{\pm i\pi /3}=\frac{1}{2}\pm \frac{\sqrt{3}}{2}i}$, the solutions tox² −x + 1 (theprimitive sixthroots of unity). The 2-cycles interchange these, as they do any points other than their fixed points, which realizes the quotient mapS₃ → S₂ by the group action on these two points. That is, the subgroupC₃ < S₃ consisting of the identity and the 3-cycles,{(), (0 1 ∞), (0 ∞ 1)}, 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 (itsSylow 2-subgroups) by conjugation and realizes the isomorphism with the group ofinner automorphisms,S₃~→ Inn(S₃) ≅ S₃.

Geometrically, this can be visualized as therotation group of thetriangular bipyramid, which is isomorphic to thedihedral group of the triangleD₃ ≅ S₃; seeanharmonic group.

Over the real and complex numbers, the topology of PGL and PSL can be determined from thefiber bundles that define them:

via thelong exact sequence of a fibration.

For both the reals and complexes, SL is acovering space of PSL, with number of sheets equal to the number ofnth roots inK; thus in particular all their higherhomotopy groups agree. For the reals, SL is a 2-fold cover of PSL forn even, and is a 1-fold cover forn odd, i.e., an isomorphism:

{±1} → SL(2n,R) → PSL(2n,R)

SL(2n + 1,R)~→ PSL(2n + 1,R)

For the complexes, SL is ann-fold cover of PSL.

For PGL, for the reals, the fiber isR^× ≅ {±1}, 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 isC^× ≅S¹, so up to homotopy,GL → PGL is acircle bundle. The higher homotopy groups of the circle vanish, so the homotopy groups ofGL(n,C) andPGL(n,C) agree forn ≥ 3. In fact,π₂ always vanishes for Lie groups, so the homotopy groups agree forn ≥ 2. Forn = 1, we have thatπ₁(GL(n,C)) =π₁(S¹) =Z. The fundamental group ofPGL(2,C) is a finite cyclic group of order 2.

Over the real and complex numbers, the projective special linear groups are theminimal (centerless)Lie group realizations for the special linear Lie algebra${\displaystyle \mathfrak{s}\mathfrak{l}(n):}$ every connected Lie group whose Lie algebra is${\displaystyle \mathfrak{s}\mathfrak{l}(n)}$ is a cover ofPSL(n,F). Conversely, itsuniversal covering group is themaximal (simply connected) element, and the intermediary realizations form alattice of covering groups.

For example,SL(2,R) has center {±1} and fundamental groupZ, and thus has universal coverSL(2,R) and covers the centerlessPSL(2,R).

Agroup homomorphismG → PGL(V) from a groupG to a projective linear group is called aprojective representation of the groupG, by analogy with alinear representation (a homomorphismG → GL(V)). These were studied byIssai Schur, who showed thatprojective representations ofG can be classified in terms oflinear representations ofcentral extensions ofG. This led to theSchur multiplier, which is used to address this question.

The projective linear group is mostly studied forn ≥ 2, though it can be defined for low dimensions.

Forn = 0 (or in factn < 0) the projective space ofK⁰ is empty, as there are no 1-dimensional subspaces of a 0-dimensional space. Thus,PGL(0,K) is the trivial group, consisting of the unique empty map from theempty set to itself. Further, the action of scalars on a 0-dimensional space is trivial, so the mapK^× → GL(0,K) is trivial, rather than an inclusion as it is in higher dimensions.

Forn = 1, the projective space ofK¹ is a single point, as there is a single 1-dimensional subspace. Thus,PGL(1,K) is the trivial group, consisting of the unique map from asingleton set to itself. Further, the general linear group of a 1-dimensional space is exactly the scalars, so the mapK^×~→ GL(1,K) is an isomorphism, corresponding toPGL(1,K) := GL(1,K) / K^× ≅ {1} being trivial.

Forn = 2,PGL(2,K) is non-trivial, but is unusual in that it is 3-transitive, unlike higher dimensions when it is only 2-transitive.

• PSL(2, 7)
• Modular group,PSL(2,Z)
• PSL(2,R)
• Möbius group,PGL(2,C) = PSL(2,C)

• 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

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

• Projective semilinear group, PΓL, which allowsfield automorphisms.
• Cremona group,Cr(Pⁿ(k)) ofbirational automorphisms; anybiregular automorphism is linear, so PGL coincides with the group of biregular automorphisms.

• Projective transformation
• Unit

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