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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, especially in the group theoretic area of
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...
, the projective linear group (also known as the projective general linear group or PGL) is the induced
action Action may refer to: * Action (narrative), a literary mode * Action fiction, a type of genre fiction * Action game, a genre of video game Film * Action film, a genre of film * ''Action'' (1921 film), a film by John Ford * ''Action'' (1980 fil ...
of the
general linear group In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
of a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
''V'' on the associated
projective space In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
P(''V''). Explicitly, the projective linear group is the
quotient group A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored" out). For examp ...
:PGL(''V'') = GL(''V'')/Z(''V'') where GL(''V'') is the
general linear group In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
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 Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learnin ...
of the action, and the notation "Z" reflects that the scalar transformations form the
center Center or centre may refer to: Mathematics *Center (geometry), the middle of an object * Center (algebra), used in various contexts ** Center (group theory) ** Center (ring theory) * Graph center, the set of all vertices of minimum eccentrici ...
of the general linear group. The projective special linear group, PSL, is defined analogously, as the induced action of the
special linear group In mathematics, the special linear group of degree ''n'' over a field ''F'' is the set of matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion. This is the normal subgroup of the genera ...
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 In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
. Here SZ is the center of SL, and is naturally identified with the group of ''n''th
roots of unity In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important in ...
in ''F'' (where ''n'' is the
dimension In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
of ''V'' and ''F'' is the base
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
). PGL and PSL are some of the fundamental groups of study, part of the so-called
classical groups In mathematics, the classical groups are defined as the special linear groups over the reals , the complex numbers and the quaternions together with special automorphism groups of symmetric or skew-symmetric bilinear forms and Hermitian or ...
, and an element of PGL is called projective linear transformation, projective transformation or
homography In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive. It is a bijection that maps lines to lines, and thus a collineation. In general, ...
. 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 In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
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 Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
, with an important example being the
modular group In mathematics, the modular group is the projective special linear group of matrices with integer coefficients and determinant 1. The matrices and are identified. The modular group acts on the upper-half of the complex plane by fractional l ...
, .


Name

The name comes from
projective geometry In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, pro ...
, where the projective group acting on
homogeneous coordinates In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometry. T ...
(''x''0:''x''1: ... :''xn'') is the underlying group of the geometry.This is therefore PGL(''n'' + 1, ''F'') for
projective space In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
of dimension ''n''
Stated differently, the natural
action Action may refer to: * Action (narrative), a literary mode * Action fiction, a type of genre fiction * Action game, a genre of video game Film * Action film, a genre of film * ''Action'' (1921 film), a film by John Ford * ''Action'' (1980 fil ...
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 PGL(2, C) of
Möbius transformation In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form f(z) = \frac of one complex variable ''z''; here the coefficients ''a'', ''b'', ''c'', ''d'' are complex numbers satisfying ''ad'' ...
s (sometimes called the
Möbius group Moebius, Möbius or Mobius may refer to: People * August Ferdinand Möbius (1790–1868), German mathematician and astronomer * Theodor Möbius (1821–1890), German philologist * Karl Möbius (1825–1908), German zoologist and ecologist * Paul ...
), which acts on the
projective line In mathematics, a projective line is, roughly speaking, the extension of a usual line by a point called a ''point at infinity''. The statement and the proof of many theorems of geometry are simplified by the resultant elimination of special cases; ...
. 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: PGL(''n'', ''F'') is the group associated to GL(''n'', ''F''), and is the projective linear group of (''n''−1)-dimensional projective space, not ''n''-dimensional projective space.


Collineations

A related group is the
collineation group In projective geometry, a collineation is a one-to-one and onto map (a bijection) from one projective space to another, or from a projective space to itself, such that the images of collinear points are themselves collinear. A collineation is th ...
, which is defined axiomatically. A collineation is an invertible (or more generally one-to-one) map which sends
collinear points In geometry, collinearity of a set of points is the property of their lying on a single line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, the term has been used for aligned o ...
to collinear points. One can define a projective space axiomatically in terms of an
incidence structure In mathematics, an incidence structure is an abstract system consisting of two types of objects and a single relationship between these types of objects. Consider the points and lines of the Euclidean plane as the two types of objects and ignore al ...
(a set of points ''P,'' lines ''L,'' and an
incidence relation In mathematics, an incidence matrix is a logical matrix that shows the relationship between two classes of objects, usually called an incidence relation. If the first class is ''X'' and the second is ''Y'', the matrix has one row for each element ...
''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 (''p'', ''l'') ∈ ''I'' then (''f''(''p''), ''g''(''l'')) ∈ ''I''. 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 ''n'' = 2 (a projective line), all points are collinear, so the collineation group is exactly the
symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group \m ...
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 ''n'' ≥ 3, the collineation group is the
projective semilinear group In linear algebra, particularly projective geometry, a semilinear map between vector spaces ''V'' and ''W'' over a field ''K'' is a function that is a linear map "up to a twist", hence ''semi''-linear, where "twist" means "field automorphism of ''K' ...
, PΓL – this is PGL, twisted by
field automorphism In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms ...
s; formally, PΓL ≅ PGL ⋊ Gal(''K''/''k''), where ''k'' is the
prime field In mathematics, the characteristic of a ring , often denoted , is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0). If this sum never reaches the additive ide ...
for ''K;'' this is the
fundamental theorem of projective geometry In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive. It is a bijection that maps lines to lines, and thus a collineation. In general, ...
. Thus for ''K'' a prime field (F''p'' or Q), we have PGL = PΓL, but for ''K'' a field with non-trivial Galois automorphisms (such as \mathbf_ for ''n'' ≥ 2 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 PΓL/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 projective ''linear'' transform. However, with the exception of the
non-Desarguesian plane In mathematics, a non-Desarguesian plane is a projective plane that does not satisfy Desargues' theorem (named after Girard Desargues), or in other words a plane that is not a Desarguesian plane. The theorem of Desargues is true in all projective s ...
s, all projective spaces are the projectivization of a linear space over a
division ring In algebra, a division ring, also called a skew field, is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicative inverse, that is, an element ...
though, as noted above, there are multiple choices of linear structure, namely a
torsor In mathematics, a principal homogeneous space, or torsor, for a group ''G'' is a homogeneous space ''X'' for ''G'' in which the stabilizer subgroup of every point is trivial. Equivalently, a principal homogeneous space for a group ''G'' is a non-e ...
over Gal(''K''/''k'') (for ''n'' ≥ 3).


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 PSO(''n''+1)), which corresponds to the
stereographic projection In mathematics, a stereographic projection is a perspective projection of the sphere, through a specific point on the sphere (the ''pole'' or ''center of projection''), onto a plane (geometry), plane (the ''projection plane'') perpendicular to ...
of rotations of the unit hypersphere, and has dimension \textstyle. 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 \textstyle.), 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 In projective geometry, a collineation is a one-to-one and onto map (a bijection) from one projective space to another, or from a projective space to itself, such that the images of collinear points are themselves collinear. A collineation is th ...
, which is instead either PΓL (for ''n'' > 2) or the full
symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group \m ...
for ''n'' = 2 (the projective line). * Every ( biregular) algebraic automorphism of a projective space is projective linear. The
birational automorphism In mathematics, birational geometry is a field of algebraic geometry in which the goal is to determine when two algebraic varieties are isomorphic outside lower-dimensional subsets. This amounts to studying mappings that are given by rational ...
s form a larger group, the
Cremona group In algebraic geometry, the Cremona group, introduced by , is the group of birational automorphisms of the n-dimensional projective space over a field It is denoted by Cr(\mathbb^n(k)) or Bir(\mathbb^n(k)) or Cr_n(k). The Cremona group is natura ...
. * 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 In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts are ...
, and GL acts transitively on ''k''-element sets of linearly independent vectors. * PGL(2, ''K'') acts sharply 3-transitively on the projective line. *:3 arbitrary points are conventionally mapped to
, 1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline (t ...
, 1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline (t ...
, 0 in alternative notation, 0, 1, ∞. In fractional linear transformation notation, the function \frac\cdot \frac maps ''a'' ↦ 0, ''b'' ↦ 1, ''c'' ↦ ∞, and is the unique such map that does so. This is the
cross-ratio In geometry, the cross-ratio, also called the double ratio and anharmonic ratio, is a number associated with a list of four collinear points, particularly points on a projective line. Given four points ''A'', ''B'', ''C'' and ''D'' on a line, th ...
(''x'', ''b''; ''a'', ''c'') – see cross-ratio: transformational approach for details. * For ''n'' ≥ 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. For ''n'' = 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 for PGL(2, 3), as P1(3) has 3+1=4 points, so 3-transitive implies 4-transitive); the invariant that is preserved is the
cross ratio In geometry, the cross-ratio, also called the double ratio and anharmonic ratio, is a number associated with a list of four collinear points, particularly points on a projective line. Given four points ''A'', ''B'', ''C'' and ''D'' on a line, the ...
, 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). * PSL(2, ''q'') and PGL(2, ''q'') (for ''q'' > 2, and ''q'' odd for PSL) are two of the four families of
Zassenhaus group In mathematics, a Zassenhaus group, named after Hans Zassenhaus, is a certain sort of doubly transitive permutation group very closely related to rank-1 groups of Lie type. Definition A Zassenhaus group is a permutation group ''G'' on a finite ...
s. * PGL(''n'', ''K'') is an
algebraic group In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Man ...
of dimension ''n''2−1 and an open subgroup of the projective space P''n''2−1. As defined, the functor PSL(''n'',''K'') does not define an algebraic group, or even an fppf sheaf, and its sheafification in the
fppf topology In mathematics, the flat topology is a Grothendieck topology used in algebraic geometry. It is used to define the theory of flat cohomology; it also plays a fundamental role in the theory of descent (faithfully flat descent). The term ''flat'' her ...
is in fact PGL(''n'',''K''). * PSL and PGL are centerless – this is because the diagonal matrices are not only the center, but also the
hypercenter In mathematics, especially in the fields of group theory and Lie theory, a central series is a kind of normal series of subgroups or Lie subalgebras, expressing the idea that the commutator is nearly trivial. For groups, the existence of a centra ...
(the quotient of a group by its center is not necessarily centerless).For PSL (except PSL(2, 2) and PSL(2, 3)) this follows by
Grün's lemma In mathematics, more specifically in group theory, a group is said to be perfect if it equals its own commutator subgroup, or equivalently, if the group has no non-trivial abelian quotients (equivalently, its abelianization, which is the univers ...
because SL is a
perfect group In mathematics, more specifically in group theory, a group is said to be perfect if it equals its own commutator subgroup, or equivalently, if the group has no non-trivial abelian quotients (equivalently, its abelianization, which is the universa ...
(hence center equals hypercenter), but for PGL and the two exceptional PSLs this requires additional checking.


Fractional linear transformations

As for
Möbius transformation In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form f(z) = \frac of one complex variable ''z''; here the coefficients ''a'', ''b'', ''c'', ''d'' are complex numbers satisfying ''ad'' ...
s, the group PGL(2, ''K'') can be interpreted as
fractional linear transformation In mathematics, a linear fractional transformation is, roughly speaking, a transformation of the form :z \mapsto \frac , which has an inverse. The precise definition depends on the nature of , and . In other words, a linear fractional transfor ...
s 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 'z'', 1 Then when ''ad''– ''bc'' ≠ 0, the action of PGL(2, ''K'') 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 In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the s ...
can be used for the group product in PGL(2, ''K'').


Finite fields

The projective special linear groups PSL(''n'', F''q'') for a
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
F''q'' are often written as PSL(''n'', ''q'') or ''Ln''(''q''). They are
finite simple group Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marked ...
s whenever ''n'' is at least 2, with two exceptions: ''L''2(2), which is isomorphic to ''S''3, the
symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group \m ...
on 3 letters, and is solvable; and ''L''2(3), which is isomorphic to ''A''4, the
alternating group In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on a set of elements is called the alternating group of degree , or the alternating group on letters and denoted by or Basic prop ...
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(''n'', ''q'') are thus
quasisimple In mathematics, a quasisimple group (also known as a covering group) is a group that is a perfect central extension ''E'' of a simple group ''S''. In other words, there is a short exact sequence :1 \to Z(E) \to E \to S \to 1 such that E = , E/ ...
: perfect central extensions of a simple group (unless ''n'' = 2 and ''q'' = 2 or 3).


History

The groups PSL(2, ''p'') were constructed by
Évariste Galois Évariste Galois (; ; 25 October 1811 – 31 May 1832) was a French mathematician and political activist. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, ...
in the 1830s, and were the second family of finite
simple group SIMPLE Group Limited is a conglomeration of separately run companies that each has its core area in International Consulting. The core business areas are Legal Services, Fiduciary Activities, Banking Intermediation and Corporate Service. The d ...
s, after the
alternating group In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on a set of elements is called the alternating group of degree , or the alternating group on letters and denoted by or Basic prop ...
s. 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(ν, ''p''), in studying the Galois group of the general equation of degree ''p''ν. The groups PSL(''n'', ''q'') (general ''n'', general finite field) were then constructed in the classic 1870 text by
Camille Jordan Marie Ennemond Camille Jordan (; 5 January 1838 – 22 January 1922) was a French mathematician, known both for his foundational work in group theory and for his influential ''Cours d'analyse''. Biography Jordan was born in Lyon and educated at ...
, '' Traité des substitutions et des équations algébriques.''


Order

The order of PGL(''n'', ''q'') is :(''q''''n'' − 1)(''qn'' − ''q'')(''qn'' − ''q''2) ⋅⋅⋅ (''q''''n'' − ''q''''n''−1)/(''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 Big ''O'' notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by Paul Bachmann, Edmund Lan ...
, 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 ''n''th root, or equivalently, dividing by the number of ''n''th
roots of unity In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important in ...
in F''q''.These are equal because they are the kernel and cokernel of the endomorphism F^ \overset F^; 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 :''L''2(2) ≅ ''S''3, ''L''2(3) ≅ ''A''4, and PGL(2, 3) ≅ ''S''4, 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) \cong A_5 :L_2(5) \cong A_5 (see
here Here is an adverb that means "in, on, or at this place". It may also refer to: Software * Here Technologies, a mapping company * Here WeGo (formerly Here Maps), a mobile app and map website by Here Technologies, Here Television * Here TV (form ...
for a proof) :L_2(9) \cong A_6 :L_4(2) \cong A_8. The isomorphism ''L''2(9) ≅ ''A''6 allows one to see the exotic outer automorphism of ''A''6 in terms of
field automorphism In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms ...
and matrix operations. The isomorphism ''L''4(2) ≅ ''A''8 is of interest in the structure of the Mathieu group M24. The associated extensions SL(''n'', ''q'') → PSL(''n'', ''q'') are covering groups of the alternating groups (
universal perfect central extension In mathematical group theory, the Schur multiplier or Schur multiplicator is the second homology group H_2(G, \Z) of a group ''G''. It was introduced by in his work on projective representations. Examples and properties The Schur multiplier \oper ...
s) for ''A''4, ''A''5, by uniqueness of the universal perfect central extension; for ''L''2(9) ≅ ''A''6, the associated extension is a perfect central extension, but not universal: there is a 3-fold
covering group In mathematics, a covering group of a topological group ''H'' is a covering space ''G'' of ''H'' such that ''G'' is a topological group and the covering map is a continuous group homomorphism. The map ''p'' is called the covering homomorphism. ...
. The groups over F5 have a number of exceptional isomorphisms: :PSL(2, 5) ≅ ''A''5 ≅ ''I'', the alternating group on five elements, or equivalently the
icosahedral group In mathematics, and especially in geometry, an object has icosahedral symmetry if it has the same symmetries as a regular icosahedron. Examples of other polyhedra with icosahedral symmetry include the regular dodecahedron (the dual of the ...
; :PGL(2, 5) ≅ ''S''5, the
symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group \m ...
on five elements; :SL(2, 5) ≅ 2 ⋅ ''A''5 ≅ 2''I'' the double cover of the alternating group ''A''5, or equivalently the
binary icosahedral group In mathematics, the binary icosahedral group 2''I'' or Coxeter&Moser: Generators and Relations for discrete groups: : Rl = Sm = Tn = RST is a certain nonabelian group of order 120. It is an extension of the icosahedral group ''I'' or (2,3,5) of o ...
. They can also be used to give a construction of an exotic map ''S''5 → ''S''6, as described below. Note however that GL(2, 5) is not a double cover of ''S''5, but is rather a 4-fold cover. A further isomorphism is: :''L''2(7) ≅ ''L''3(2) is the simple group of order 168, the second-smallest non-abelian simple group, and is not an alternating group; see
PSL(2,7) In mathematics, the projective special linear group , isomorphic to , is a finite simple group that has important applications in algebra, geometry, and number theory. It is the automorphism group of the Klein quartic as well as the symmetry group ...
. 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 In mathematics, the projective unitary group is the quotient of the unitary group by the right multiplication of its center, , embedded as scalars. Abstractly, it is the holomorphic isometry group of complex projective space, just as the projectiv ...
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(''n'', ''q'') acts on the projective space P''n''−1(''q''), which has (''q''''n''−1)/(''q''−1) points, and this yields a map from the projective linear group to the symmetric group on (''q''''n''−1)/(''q''−1) points. For ''n'' = 2, this is the projective line P1(''q'') which has (''q''2−1)/(''q''−1) = ''q''+1 points, so there is a map PGL(2, ''q'') → ''S''''q''+1. To understand these maps, it is useful to recall these facts: * The order of PGL(2, ''q'') is ::(q^2-1)(q^2-q)/(q-1)=q^3-q=(q-1)q(q+1); :the order of PSL(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) → ''S''3, of order 6, which is an isomorphism. ** The inverse map (a projective representation of ''S''3) can be realized by the
anharmonic group In geometry, the cross-ratio, also called the double ratio and anharmonic ratio, is a number associated with a list of four collinear points, particularly points on a projective line. Given four points ''A'', ''B'', ''C'' and ''D'' on a line, the ...
, and more generally yields an embedding ''S''3 → PGL(2, ''q'') for all fields. *PSL(2, 3) < PGL(2, 3) → ''S''4, of orders 12 and 24, the latter of which is an isomorphism, with PSL(2, 3) being the alternating group. ** The anharmonic group gives a partial map in the opposite direction, mapping ''S''3 → PGL(2, 3) as the stabilizer of the point −1. *PSL(2, 4) = PGL(2, 4) → ''S''5, of order 60, yielding the alternating group ''A''5. *PSL(2, 5) < PGL(2, 5) → ''S''6, of orders 60 and 120, which yields an embedding of ''S''5 (respectively, ''A''5) as a ''transitive'' subgroup of ''S''6 (respectively, ''A''6). This is an example of an exotic map ''S''5 → ''S''6, and can be used to construct the exceptional outer automorphism of ''S''6. Note that the isomorphism PGL(2, 5) ≅ ''S''5 is not transparent from this presentation: there is no particularly natural set of 5 elements on which PGL(2, 5) acts.


Action on ''p'' points

While PSL(''n'', ''q'') naturally acts on (''q''''n''−1)/(''q''−1) = 1+''q''+...+''q''''n''−1 points, non-trivial actions on fewer points are rarer. Indeed, for PSL(2, ''p'') 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.Since ''p'' divides the order of the group, the group does not embed in (or, since simple, map non-trivially to) ''Sk'' for ''k'' < ''p'', as ''p'' does not divide the order of this latter group. This was first observed by
Évariste Galois Évariste Galois (; ; 25 October 1811 – 31 May 1832) was a French mathematician and political activist. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, ...
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 ''Sp''. In all but the last case, PSL(2, 11), it corresponds to an exceptional isomorphism, where the right-most group has an obvious action on ''p'' points: * L_2(2) \cong S_3 \twoheadrightarrow S_2 via the sign map; * L_2(3) \cong A_4 \twoheadrightarrow A_3 \cong C_3 via the quotient by the Klein 4-group; * L_2(5) \cong A_5. To construct such an isomorphism, one needs to consider the group ''L''2(5) as a Galois group of a Galois cover ''a''5: ''X''(5) → ''X''(1) = P1, where ''X''(''N'') is a
modular curve In number theory and algebraic geometry, a modular curve ''Y''(Γ) is a Riemann surface, or the corresponding algebraic curve, constructed as a quotient of the complex upper half-plane H by the action of a congruence subgroup Γ of the modular ...
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. *''L''2(7) ≅ ''L''3(2) which acts on the 1+2+4 = 7 points of the
Fano plane In finite geometry, the Fano plane (after Gino Fano) is a finite projective plane with the smallest possible number of points and lines: 7 points and 7 lines, with 3 points on every line and 3 lines through every point. These points and lines c ...
(projective plane over F2); this can also be seen as the action on order 2
biplane A biplane is a fixed-wing aircraft with two main wings stacked one above the other. The first powered, controlled aeroplane to fly, the Wright Flyer, used a biplane wing arrangement, as did many aircraft in the early years of aviation. While ...
, 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 In mathematics, the ADE classification (originally ''A-D-E'' classifications) is a situation where certain kinds of objects are in correspondence with simply laced Dynkin diagrams. The question of giving a common origin to these classifications, r ...
: these actions correspond to products (as sets, not as groups) of the groups as ''A''4 × Z/5Z, ''S''4 × Z/7Z, and ''A''5 × Z/11Z, where the groups ''A''4, ''S''4 and ''A''5 are the isometry groups of the
Platonic solid In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edges c ...
s, and correspond to ''E''6, ''E''7, and ''E''8 under the
McKay correspondence In mathematics, the McKay graph of a finite-dimensional representation of a finite group is a weighted quiver encoding the structure of the representation theory of . Each node represents an irreducible representation of . If are irreducibl ...
. These three exceptional cases are also realized as the geometries of polyhedra (equivalently, tilings of Riemann surfaces), respectively: the
compound of five tetrahedra The polyhedral compound, compound of five tetrahedron, tetrahedra is one of the five regular polyhedral compounds. This Polyhedral compound, compound polyhedron is also a stellation of the regular icosahedron. It was first described by Edmund Hes ...
inside the icosahedron (sphere, genus 0), the order 2 biplane (complementary
Fano plane In finite geometry, the Fano plane (after Gino Fano) is a finite projective plane with the smallest possible number of points and lines: 7 points and 7 lines, with 3 points on every line and 3 lines through every point. These points and lines c ...
) inside the Klein quartic (genus 3), and the order 3 biplane (
Paley biplane In combinatorial mathematics, a block design is an incidence structure consisting of a set together with a family of subsets known as ''blocks'', chosen such that frequency of the elements satisfies certain conditions making the collection of bl ...
) inside the
buckyball surface In mathematics, the ADE classification (originally ''A-D-E'' classifications) is a situation where certain kinds of objects are in correspondence with simply laced Dynkin diagrams. The question of giving a common origin to these classifications, ...
(genus 70). The action of ''L''2(11) can be seen algebraically as due to an exceptional inclusion L_2(5) \hookrightarrow L_2(11) – 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 ''S''4, and this also has a biplane geometry.) Geometrically, this action can be understood via a ''
biplane geometry In combinatorial mathematics, a block design is an incidence structure consisting of a set together with a family of subsets known as ''blocks'', chosen such that frequency of the elements satisfies certain conditions making the collection of bl ...
,'' which is defined as follows. A biplane geometry is a
symmetric design In combinatorial mathematics, a block design is an incidence structure consisting of a set together with a family of subsets known as ''blocks'', chosen such that frequency of the elements satisfies certain conditions making the collection of bl ...
(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 In combinatorial mathematics, a block design is an incidence structure consisting of a set together with a family of subsets known as ''blocks'', chosen such that frequency of the elements satisfies certain conditions making the collection of bl ...
, obtained from the
Paley digraph In mathematics, Paley graphs are dense undirected graphs constructed from the members of a suitable finite field by connecting pairs of elements that differ by a quadratic residue. The Paley graphs form an infinite family of conference graphs, w ...
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 ''S''11 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 660/11 = 60 (and on which the icosahedral group acts) naturally has the structure of a buckeyball, which is used in the construction of the
buckyball surface In mathematics, the ADE classification (originally ''A-D-E'' classifications) is a situation where certain kinds of objects are in correspondence with simply laced Dynkin diagrams. The question of giving a common origin to these classifications, ...
.


Mathieu groups

The group PSL(3, 4) can be used to construct the
Mathieu group In group theory, a topic in abstract algebra, the Mathieu groups are the five sporadic simple groups ''M''11, ''M''12, ''M''22, ''M''23 and ''M''24 introduced by . They are multiply transitive permutation groups on 11, 12, 22, 23 or 24 obje ...
M24, one of the sporadic simple groups; in this context, one refers to PSL(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 a
Steiner system 250px, thumbnail, The Fano plane is a Steiner triple system S(2,3,7). The blocks are the 7 lines, each containing 3 points. Every pair of points belongs to a unique line. In combinatorial mathematics, a Steiner system (named after Jakob Steiner) ...
of type S(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 which PSL(3, 4) 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 PGL(3, 4), then to the
projective semilinear group In linear algebra, particularly projective geometry, a semilinear map between vector spaces ''V'' and ''W'' over a field ''K'' is a function that is a linear map "up to a twist", hence ''semi''-linear, where "twist" means "field automorphism of ''K' ...
PΓL(3, 4), and finally to the Mathieu group M24. M24 also contains copies of PSL(2, 11), which is maximal in M22, and PSL(2, 23), which is maximal in M24, and can be used to construct M24.Conway, Sloane, SPLAG


Hurwitz surfaces

PSL groups arise as
Hurwitz group In mathematics, Hurwitz's automorphisms theorem bounds the order of the group of automorphisms, via orientation-preserving conformal mappings, of a compact Riemann surface of genus ''g'' > 1, stating that the number of such automorphisms ...
s (automorphism groups of
Hurwitz surface In Riemann surface theory and hyperbolic geometry, a Hurwitz surface, named after Adolf Hurwitz, is a compact Riemann surface with precisely 84(''g'' − 1) automorphisms, where ''g'' is the genus of the surface. This number is maximal by vir ...
s – algebraic curves of maximal possibly symmetry group). The Hurwitz surface of lowest genus, the
Klein quartic In hyperbolic geometry, the Klein quartic, named after Felix Klein, is a compact Riemann surface of genus with the highest possible order automorphism group for this genus, namely order orientation-preserving automorphisms, and automorphisms ...
(genus 3), has automorphism group isomorphic to PSL(2, 7) (equivalently GL(3, 2)), while the Hurwitz surface of second-lowest genus, the
Macbeath surface In Riemann surface theory and hyperbolic geometry, the Macbeath surface, also called Macbeath's curve or the Fricke–Macbeath curve, is the genus-7 Hurwitz surface. The automorphism group of the Macbeath surface is the simple group Projective line ...
(genus 7), has automorphism group isomorphic to PSL(2, 8). In fact, many but not all simple groups arise as Hurwitz groups (including the
monster group In the area of abstract algebra known as group theory, the monster group M (also known as the Fischer–Griess monster, or the friendly giant) is the largest sporadic simple group, having order    246320597611213317192329314147 ...
, though not all alternating groups or sporadic groups), though PSL is notable for including the smallest such groups.


Modular group

The groups PSL(2, Z/''n''Z) arise in studying the
modular group In mathematics, the modular group is the projective special linear group of matrices with integer coefficients and determinant 1. The matrices and are identified. The modular group acts on the upper-half of the complex plane by fractional l ...
, PSL(2, Z), as quotients by reducing all elements mod ''n''; the kernels are called the
principal congruence subgroup In mathematics, a congruence subgroup of a matrix group with integer entries is a subgroup defined by congruence conditions on the entries. A very simple example would be invertible 2 × 2 integer matrices of determinant 1, in which the ...
s. A noteworthy subgroup of the projective ''general'' linear group PGL(2, Z) (and of the projective special linear group PSL(2, Z 'i'') is the symmetries of the set ⊂ P1(C)In projective coordinates, the points are given by :1 :1 and :0 which explains why their stabilizer is represented by integral matrices. these also occur in the six cross-ratios. The subgroup can be expressed as
fractional linear transformation In mathematics, a linear fractional transformation is, roughly speaking, a transformation of the form :z \mapsto \frac , which has an inverse. The precise definition depends on the nature of , and . In other words, a linear fractional transfor ...
s, 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 PSL(2, Z), while the bottom row is the three 2-cycles, and are in PGL(2, Z) and PSL(2, Z 'i'', but not in PSL(2, Z), hence realized either as matrices with determinant −1 and integer coefficients, or as matrices with determinant 1 and
Gaussian integer In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as \mathbf /ma ...
coefficients. This maps to the symmetries of ⊂ P1(''n'') under reduction mod ''n''. Notably, for ''n'' = 2, this subgroup maps isomorphically to PGL(2, Z/2Z) = PSL(2, Z/2Z) ≅ ''S''3,This isomorphism can be seen by removing the minus signs in matrices, which yields the matrices for PGL(2, 2) and thus provides a splitting \operatorname(2,\mathbf/2) \hookrightarrow \operatorname(2,\mathbf) for the quotient map \operatorname(2,\mathbf) \twoheadrightarrow \operatorname(2,\mathbf/2). A further property of this subgroup is that the quotient map ''S''3 → ''S''2 is realized by the group action. That is, the subgroup ''C''3 < ''S''3 consisting of the 3-cycles and the identity () (0 1 ∞) (0 ∞ 1) stabilizes the
golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( ...
and inverse golden ratio \varphi_\pm = \frac, 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 ''S''3 on the 2-cycles (its Sylow 2-subgroups) by conjugation and realizing the isomorphism S_3 \overset \operatorname(S_3) \cong S_3.


Topology

Over the real and complex numbers, the topology of PGL and PSL can be determined from the
fiber bundle In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a p ...
s that define them: :\begin \mathrm &\cong& K^* &\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 A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties. Definition Let X be a topological space. A covering of X is a continuous map : \pi : E \rightarrow X such that there exists a discrete spa ...
of PSL, with number of sheets equal to the number of ''n''th roots in ''K''; thus in particular all their higher
homotopy groups In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homo ...
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: : → SL(2''n'', R) → PSL(2''n'', R) :\operatorname(2n+1, \mathbf) \overset \operatorname(2n+1,\mathbf) 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(''n'', C) and PGL(''n'', C) 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(GL(''n'', C)) = π1(S1) = Z. The fundamental group of PGL(''2'', C) 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 PSL(''n'', ''F''). Conversely, its universal covering group is the ''maximal'' (
simply connected In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spac ...
) element, and the intermediary realizations form a
lattice of covering groups In mathematics, a covering group of a topological group ''H'' is a covering space ''G'' of ''H'' such that ''G'' is a topological group and the covering map is a continuous group homomorphism. The map ''p'' is called the covering homomorphism. A ...
. For example, SL(2, R) has center and fundamental group Z, and thus has universal cover and covers the centerless PSL(2, R).


Representation theory

A
group homomorphism In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that : h(u*v) = h(u) \cdot h(v) wh ...
''G'' → PGL(''V'') from a group ''G'' to a projective linear group is called a
projective representation In the field of representation theory in mathematics, a projective representation of a group ''G'' on a vector space ''V'' over a field ''F'' is a group homomorphism from ''G'' to the projective linear group \mathrm(V) = \mathrm(V) / F^*, where G ...
of the group ''G,'' by analogy with a
linear representation Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essenc ...
(a homomorphism G → GL(''V'')). These were studied by
Issai Schur Issai Schur (10 January 1875 – 10 January 1941) was a Russian mathematician who worked in Germany for most of his life. He studied at the University of Berlin. He obtained his doctorate in 1901, became lecturer in 1903 and, after a stay at th ...
, 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 In mathematical group theory, the Schur multiplier or Schur multiplicator is the second homology group H_2(G, \Z) of a group ''G''. It was introduced by in his work on projective representations. Examples and properties The Schur multiplier \oper ...
, 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 (or in fact ''n'' < 0) the projective space of ''K''0 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 the
empty set In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...
to itself. Further, the action of scalars on a 0-dimensional space is trivial, so the map ''K*'' → GL(0, ''K'') is trivial, rather than an inclusion as it is in higher dimensions. For ''n'' = 1, the projective space of ''K''1 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 a
singleton set In mathematics, a singleton, also known as a unit set or one-point set, is a set with exactly one element. For example, the set \ is a singleton whose single element is 0. Properties Within the framework of Zermelo–Fraenkel set theory, the ...
to itself. Further, the general linear group of a 1-dimensional space is exactly the scalars, so the map K^* \overset \operatorname(1,K) is an isomorphism, corresponding to PGL(1, ''K'') := GL(1, ''K'')/''K*'' ≅ being trivial. For ''n'' = 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.


Examples

*
PSL(2,7) In mathematics, the projective special linear group , isomorphic to , is a finite simple group that has important applications in algebra, geometry, and number theory. It is the automorphism group of the Klein quartic as well as the symmetry group ...
*
Modular group In mathematics, the modular group is the projective special linear group of matrices with integer coefficients and determinant 1. The matrices and are identified. The modular group acts on the upper-half of the complex plane by fractional l ...
, PSL(2, Z) *
PSL(2,R) In mathematics, the special linear group SL(2, R) or SL2(R) is the group of 2 × 2 real matrices with determinant one: : \mbox(2,\mathbf) = \left\. It is a connected non-compact simple real Lie group of dimension 3 with ...
*
Möbius group Moebius, Möbius or Mobius may refer to: People * August Ferdinand Möbius (1790–1868), German mathematician and astronomer * Theodor Möbius (1821–1890), German philologist * Karl Möbius (1825–1908), German zoologist and ecologist * Paul ...
, PGL(2, C) = PSL(2, C)


Subgroups

*
Projective orthogonal group In projective geometry and linear algebra, the projective orthogonal group PO is the induced action of the orthogonal group of a quadratic space ''V'' = (''V'',''Q'')A quadratic space is a vector space ''V'' together with a quadratic form ''Q''; ...
, PO –
maximal compact subgroup In mathematics, a maximal compact subgroup ''K'' of a topological group ''G'' is a subgroup ''K'' that is a compact space, in the subspace topology, and maximal amongst such subgroups. Maximal compact subgroups play an important role in the class ...
of PGL *
Projective unitary group In mathematics, the projective unitary group is the quotient group, quotient of the unitary group by the right multiplication of its centre of a group, center, , embedded as scalars. Abstractly, it is the Holomorphic function, holomorphic isometry ...
, PU *
Projective special orthogonal group In projective geometry and linear algebra, the projective orthogonal group PO is the induced action of the orthogonal group of a quadratic space ''V'' = (''V'',''Q'')A quadratic space is a vector space ''V'' together with a quadratic form ''Q''; th ...
, PSO – maximal compact subgroup of PSL *
Projective special unitary group In mathematics, the projective unitary group is the quotient of the unitary group by the right multiplication of its center, , embedded as scalars. Abstractly, it is the holomorphic isometry group of complex projective space, just as the projectiv ...
, PSU


Larger groups

The projective linear group is contained within larger groups, notably: *
Projective semilinear group In linear algebra, particularly projective geometry, a semilinear map between vector spaces ''V'' and ''W'' over a field ''K'' is a function that is a linear map "up to a twist", hence ''semi''-linear, where "twist" means "field automorphism of ''K' ...
, PΓL, which allows
field automorphism In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms ...
s. *
Cremona group In algebraic geometry, the Cremona group, introduced by , is the group of birational automorphisms of the n-dimensional projective space over a field It is denoted by Cr(\mathbb^n(k)) or Bir(\mathbb^n(k)) or Cr_n(k). The Cremona group is natura ...
, ''Cr''(P''n''(''k'')) of
birational automorphism In mathematics, birational geometry is a field of algebraic geometry in which the goal is to determine when two algebraic varieties are isomorphic outside lower-dimensional subsets. This amounts to studying mappings that are given by rational ...
s; any biregular automorphism is linear, so PGL coincides with the group of biregular automorphisms.


See also

*
Projective transformation In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive. It is a bijection that maps lines to lines, and thus a collineation. In general, ...
*
Unit Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * ''Unit'' (alb ...


Notes


References

* {{DEFAULTSORT:Projective Linear Group Lie groups Projective geometry de:Allgemeine lineare Gruppe#Projektive lineare Gruppe