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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no general consensus abo ...
, the general linear group of degree ''n'' is the set of
invertible matrices In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and t ...
, together with the operation of ordinary
matrix multiplication In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix (mathematics), matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the nu ...

matrix multiplication
. This forms a
group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible, with identity matrix as the identity element of the group. The group is so named because the columns of an invertible matrix are
linearly independent In the theory of vector spaces, a set of vectors is said to be if at least one of the vectors in the set can be defined as a linear combinationIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics ...
, hence the vectors/points they define are in
general linear position In algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zero of a function, zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commu ...
, and matrices in the general linear group take points in general linear position to points in general linear position. To be more precise, it is necessary to specify what kind of objects may appear in the entries of the matrix. For example, the general linear group over R (the set of
real numbers Real may refer to: * Reality, the state of things as they exist, rather than as they may appear or may be thought to be Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish col ...

real numbers
) is the group of invertible matrices of real numbers, and is denoted by GL''n''(R) or . More generally, the general linear group of degree ''n'' over any
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 grassl ...
''F'' (such as the
complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

complex number
s), or a ring ''R'' (such as the ring of
integer An integer (from the Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Re ...
s), is the set of invertible matrices with entries from ''F'' (or ''R''), again with matrix multiplication as the group operation.Here rings are assumed to be associative and unital. Typical notation is GL''n''(''F'') or , or simply GL(''n'') if the field is understood. More generally still, the general linear group of a vector space GL(''V'') is the abstract
automorphism group In mathematics, the automorphism group of an object ''X'' is the group (mathematics), group consisting of automorphisms of ''X''. For example, if ''X'' is a Dimension (vector space), finite-dimensional vector space, then the automorphism group of ' ...
, not necessarily written as matrices. The
special linear group In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
, written or SL''n''(''F''), is the
subgroup In group theory, a branch of mathematics, given a group (mathematics), group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely ...
of consisting of matrices with a
determinant In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

determinant
of 1. The group and its
subgroup In group theory, a branch of mathematics, given a group (mathematics), group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely ...
s are often called linear groups or matrix groups (the abstract group GL(''V'') is a linear group but not a matrix group). These groups are important in the theory of
group representation In the mathematical field of representation theory Representation theory is a branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, struc ...
s, and also arise in the study of spatial
symmetries Symmetry (from Ancient Greek, Greek συμμετρία ''symmetria'' "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" ...

symmetries
and symmetries of
vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
s in general, as well as the study of
polynomials In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
. The
modular group In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
may be realised as a quotient of the special linear group . If , then the group is not abelian.


General linear group of a vector space

If ''V'' is a
vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
over the field ''F'', the general linear group of ''V'', written GL(''V'') or Aut(''V''), is the group of all
automorphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

automorphism
s of ''V'', i.e. the set of all
bijective In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
linear transformation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s , together with functional composition as group operation. If ''V'' has finite
dimension In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...
''n'', then GL(''V'') and are
isomorphic In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
. The isomorphism is not canonical; it depends on a choice of
basis Basis may refer to: Finance and accounting *Adjusted basisIn tax accounting, adjusted basis is the net cost of an asset after adjusting for various tax-related items. Adjusted Basis or Adjusted Tax Basis refers to the original cost or other b ...
in ''V''. Given a basis of ''V'' and an automorphism ''T'' in GL(''V''), we have then for every basis vector ''e''''i'' that : T(e_i) = \sum_^n a_ e_j for some constants ''a''''ij'' in ''F''; the matrix corresponding to ''T'' is then just the matrix with entries given by the ''a''''ij''. In a similar way, for a commutative ring ''R'' the group may be interpreted as the group of automorphisms of a ''
free Free may refer to: Concept * Freedom, having the ability to act or change without constraint * Emancipate, to procure political rights, as for a disenfranchised group * Free will, control exercised by rational agents over their actions and decis ...

free
'' ''R''-module ''M'' of rank ''n''. One can also define GL(''M'') for any ''R''-module, but in general this is not isomorphic to (for any ''n'').


In terms of determinants

Over a field ''F'', a matrix is invertible if and only if its
determinant In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

determinant
is nonzero. Therefore, an alternative definition of is as the group of matrices with nonzero determinant. Over a
commutative ring In , a branch of , a commutative ring is a in which the multiplication operation is . The study of commutative rings is called . Complementarily, is the study of s where multiplication is not required to be commutative. Definition and first e ...
''R'', more care is needed: a matrix over ''R'' is invertible if and only if its determinant is a
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 (album), ...
in ''R'', that is, if its determinant is invertible in ''R''. Therefore, may be defined as the group of matrices whose determinants are units. Over a non-commutative ring ''R'', determinants are not at all well behaved. In this case, may be defined as the
unit group In the branch of abstract algebra known as ring theory, a unit of a ring (mathematics), ring R is any element u \in R that has a multiplicative inverse in R: an element v \in R such that :vu = uv = 1, where is the multiplicative identity. The s ...
of the
matrix ring In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...
.


As a Lie group


Real case

The general linear group over the field of
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s is a real
Lie group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
of dimension ''n''2. To see this, note that the set of all real matrices, M''n''(R), forms a
real vector space Real may refer to: * Reality, the state of things as they exist, rather than as they may appear or may be thought to be Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish col ...
of dimension ''n''2. The subset consists of those matrices whose
determinant In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

determinant
is non-zero. The determinant is a
polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

polynomial
map, and hence is an open affine subvariety of M''n''(R) (a
non-empty #REDIRECT Empty set #REDIRECT Empty set #REDIRECT Empty set#REDIRECT Empty set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algeb ...
open subset Open or OPEN may refer to: Music * Open (band) Open is a band. Background Drummer Pete Neville has been involved in the Sydney/Australian music scene for a number of years. He has recently completed a Masters in screen music at the Australian ...
of M''n''(R) in the
Zariski topology In algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zero of a function, zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commut ...
), and therefore a
smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's s ...
of the same dimension. The
Lie algebra In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
of , denoted \mathfrak_n, consists of all real matrices with the
commutator In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
serving as the Lie bracket. As a manifold, is not connected but rather has two connected components: the matrices with positive determinant and the ones with negative determinant. The
identity component In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
, denoted by , consists of the real matrices with positive determinant. This is also a Lie group of dimension ''n''2; it has the same Lie algebra as . The group is also noncompact. “The”
maximal compact subgroupIn 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 element, maximal amongst such subgroups. Maximal compact subgroups play an important rol ...
of is the
orthogonal group In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
O(''n''), while "the" maximal compact subgroup of is the
special orthogonal group In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
SO(''n''). As for SO(''n''), the group is not
simply connected In topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric objec ...
(except when , but rather has a
fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence class In mathematics, when the elements of some set (mathematics), set have a notion of equivalence (formalized ...

fundamental group
isomorphic to Z for or Z2 for .


Complex case

The general linear group over the field of
complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

complex number
s, , is a ''complex''
Lie group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
of complex dimension ''n''2. As a real Lie group (through realification) it has dimension 2''n''2. The set of all real matrices forms a real Lie subgroup. These correspond to the inclusions :GL(''n'', R) < GL(''n'', C) < GL(''2n'', R), which have real dimensions ''n''2, 2''n''2, and . Complex ''n''-dimensional matrices can be characterized as real 2''n''-dimensional matrices that preserve a
linear complex structureIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
— concretely, that commute with a matrix ''J'' such that , where ''J'' corresponds to multiplying by the imaginary unit ''i''. The
Lie algebra In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
corresponding to consists of all complex matrices with the
commutator In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
serving as the Lie bracket. Unlike the real case, is connected. This follows, in part, since the multiplicative group of complex numbers C is connected. The group manifold is not compact; rather its
maximal compact subgroupIn 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 element, maximal amongst such subgroups. Maximal compact subgroups play an important rol ...
is the
unitary group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
U(''n''). As for U(''n''), the group manifold is not
simply connected In topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric objec ...
but has a
fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence class In mathematics, when the elements of some set (mathematics), set have a notion of equivalence (formalized ...

fundamental group
isomorphic to Z.


Over finite fields

If ''F'' is a
finite field In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
with ''q'' elements, then we sometimes write instead of . When ''p'' is prime, is the
outer automorphism groupIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
of the group Z, and also the
automorphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

automorphism
group, because Z is abelian, so the
inner automorphism group In abstract algebra an inner automorphism is an automorphism In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry ...
is trivial. The order of is: : \prod_^(q^n-q^k)=(q^n - 1)(q^n - q)(q^n - q^2)\ \cdots\ (q^n - q^). This can be shown by counting the possible columns of the matrix: the first column can be anything but the zero vector; the second column can be anything but the multiples of the first column; and in general, the ''k''th column can be any vector not in the
linear span In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
of the first columns. In ''q''-analog notation, this is q!(q-1)^n q^. For example, has order . It is the automorphism group of the
Fano plane In finite geometry, the Fano plane (after Gino Fano) is the Projective plane#Finite projective planes, finite projective plane of order 2. It is the finite projective plane with the smallest possible number of points and lines: 7 points and 7 li ...

Fano plane
and of the group Z, and is also known as . More generally, one can count points of
Grassmannian In mathematics, the Grassmannian is a space that parameterizes all -Dimension, dimensional linear subspaces of the -dimensional vector space . For example, the Grassmannian is the space of lines through the origin in , so it is the same as the p ...
over ''F'': in other words the number of subspaces of a given dimension ''k''. This requires only finding the order of the
stabilizer Stabilizer, stabiliser, stabilisation or stabilization may refer to: Chemistry and food processing * Stabilizer (chemistry), a substance added to prevent unwanted change in state of another substance ** Polymer stabilizers are stabilizers used s ...
subgroup of one such subspace and dividing into the formula just given, by the orbit-stabilizer theorem. These formulas are connected to the Schubert decomposition of the Grassmannian, and are ''q''-analogs of the
Betti number In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of ''n''-dimensional simplicial complexes. For the most reasonable finite-dimensional spaces (such as compact manifolds, finite simpl ...
s of complex Grassmannians. This was one of the clues leading to the
Weil conjectures In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. Note that in the limit the order of goes to 0! – but under the correct procedure (dividing by ) we see that it is the order of the symmetric group (See Lorscheid's article) – in the philosophy of the
field with one element In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
, one thus interprets the
symmetric group In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathemati ...
as the general linear group over the field with one element: .


History

The general linear group over a prime field, , was constructed and its order computed by
Évariste Galois Évariste Galois (; ; 25 October 1811 – 31 May 1832) was a French mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ...
in 1832, in his last letter (to Chevalier) and second (of three) attached manuscripts, which he used in the context of studying the
Galois group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
of the general equation of order ''p''''ν''.


Special linear group

The special linear group, , is the group of all matrices with
determinant In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

determinant
1. They are special in that they lie on a
subvariety A subvariety (Latin: ''subvarietas'') in botanical nomenclature Botanical nomenclature is the formal, scientific naming of plants. It is related to, but distinct from Alpha taxonomy, taxonomy. Plant taxonomy is concerned with grouping and classif ...
– they satisfy a polynomial equation (as the determinant is a polynomial in the entries). Matrices of this type form a group as the determinant of the product of two matrices is the product of the determinants of each matrix. is a
normal subgroup In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...
of . If we write ''F''× for the
multiplicative group In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
of ''F'' (excluding 0), then the determinant is a
group homomorphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

group homomorphism
:det: GL(''n'', ''F'') → ''F''×. that is surjective and its
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 ...
is the special linear group. Therefore, by the
first isomorphism theorem In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist for ...

first isomorphism theorem
, is
isomorphic In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

isomorphic
to ''F''×. In fact, can be written as a
semidirect product In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
: :GL(''n'', ''F'') = SL(''n'', ''F'') ⋊ ''F''× The special linear group is also the derived group (also known as commutator subgroup) of the GL(''n'', ''F'') (for a field or a
division ringIn algebra, a division ring, also called a skew field, is a ring (mathematics), ring in which division (mathematics), division is possible. Specifically, it is a zero ring, nonzero ring in which every nonzero element has a multiplicative inverse, th ...
''F'') provided that n \ne 2 or ''k'' is not the field with two elements., Theorem II.9.4 When ''F'' is R or C, is a Lie subgroup of of dimension . The
Lie algebra In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
of consists of all matrices over ''F'' with vanishing
trace Trace may refer to: Arts and entertainment Music * ''Trace'' (Son Volt album), 1995 * ''Trace'' (Died Pretty album), 1993 * Trace (band) Trace was a Netherlands, Dutch progressive rock trio founded by Rick van der Linden in 1974 after leavin ...
. The Lie bracket is given by the
commutator In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
. The special linear group can be characterized as the group of ''
volume Volume is a scalar quantity expressing the amount Quantity or amount is a property that can exist as a multitude Multitude is a term for a group of people who cannot be classed under any other distinct category, except for their shared fact ...

volume
and orientation-preserving'' linear transformations of R''n''. The group is simply connected, while is not. has the same fundamental group as , that is, Z for and Z2 for .


Other subgroups


Diagonal subgroups

The set of all invertible
diagonal matrices In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. An example of a 2×2 diagonal matrix is \left begin 3 & 0 \\ 0 & 2 \end\right/math>, while ...
forms a subgroup of isomorphic to (''F''×)''n''. In fields like R and C, these correspond to rescaling the space; the so-called dilations and contractions. A scalar matrix is a diagonal matrix which is a constant times the
identity matrix In linear algebra, the identity matrix of size ''n'' is the ''n'' × ''n'' square matrix In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structu ...

identity matrix
. The set of all nonzero scalar matrices forms a subgroup of isomorphic to ''F''×. This group is the
center Center or centre may refer to: Mathematics *Center (geometry) In geometry, a centre (or center) (from Ancient Greek language, Greek ''κέντρον'') of an object is a point in some sense in the middle of the object. According to the speci ...
of . In particular, it is a normal, abelian subgroup. The center of is simply the set of all scalar matrices with unit determinant, and is isomorphic to the group of ''n''th
roots of unity In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis) ...
in the field ''F''.


Classical groups

The so-called
classical group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
s are subgroups of GL(''V'') which preserve some sort of
bilinear form In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
on a vector space ''V''. These include the *
orthogonal group In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, O(''V''), which preserves a
non-degenerate In mathematics, specifically linear algebra, a degenerate bilinear form on a vector space ''V'' is a bilinear form such that the map from ''V'' to ''V''∗ (the dual space of ''V'') given by is not an isomorphism. An equivalent definition when ...
quadratic form In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
on ''V'', *
symplectic group In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical Group (mathematics), groups, denoted and for positive integer ''n'' and field (mathematics), field F (usually C or R). The lat ...
, Sp(''V''), which preserves a
symplectic form In mathematics, a symplectic vector space is a vector space ''V'' over a Field (mathematics), field ''F'' (for example the real numbers R) equipped with a symplectic bilinear form. A symplectic bilinear form is a map (mathematics), mapping that is ...
on ''V'' (a non-degenerate
alternating form In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
), *
unitary group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
, U(''V''), which, when , preserves a non-degenerate
hermitian form In mathematics, a sesquilinear form is a generalization of a bilinear form that, in turn, is a generalization of the concept of the dot product of Euclidean space. A bilinear form is linear map, linear in each of its arguments, but a sesquilinear fo ...
on ''V''. These groups provide important examples of Lie groups.


Related groups and monoids


Projective linear group

The
projective linear group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
and the
projective special linear group In mathematics, especially in the group theory, group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced Group action (mathematics), action of the general linear group ...
are the quotients of and by their centers (which consist of the multiples of the identity matrix therein); they are the induced Group action (mathematics), action on the associated projective space.


Affine group

The affine group is an group extension, extension of by the group of translations in ''F''''n''. It can be written as a
semidirect product In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
: :Aff(''n'', ''F'') = GL(''n'', ''F'') ⋉ ''F''''n'' where acts on ''F''''n'' in the natural manner. The affine group can be viewed as the group of all affine transformations of the affine space underlying the vector space ''F''''n''. One has analogous constructions for other subgroups of the general linear group: for instance, the special affine group is the subgroup defined by the semidirect product, , and the Poincaré group is the affine group associated to the Lorentz group, .


General semilinear group

The general semilinear group is the group of all invertible semilinear transformations, and contains GL. A semilinear transformation is a transformation which is linear “up to a twist”, meaning “up to a field automorphism under scalar multiplication”. It can be written as a semidirect product: :ΓL(''n'', ''F'') = Gal(''F'') ⋉ GL(''n'', ''F'') where Gal(''F'') is the
Galois group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
of ''F'' (over its prime field), which acts on by the Galois action on the entries. The main interest of is that the associated projective semilinear group (which contains is the collineation group of projective space, for , and thus semilinear maps are of interest in projective geometry.


Full linear monoid

If one removes the restriction of the determinant being non-zero, the resulting algebraic structure is a monoid, usually called the full linear monoid, but occasionally also ''full linear semigroup'', ''general linear monoid'' etc. It is actually a regular semigroup.


Infinite general linear group

The infinite general linear group or direct limit of groups, stable general linear group is the direct limit of the inclusions as the upper left block matrix. It is denoted by either GL(''F'') or , and can also be interpreted as invertible infinite matrices which differ from the identity matrix in only finitely many places. It is used in algebraic K-theory to define Algebraic K-theory#K1, K1, and over the reals has a well-understood topology, thanks to Bott periodicity. It should not be confused with the space of (bounded) invertible operators on a Hilbert space, which is a larger group, and topologically much simpler, namely contractible – see Kuiper's theorem.


See also

* List of finite simple groups * SL2(R), SL2(R) * Representation theory of SL2(R), Representation theory of SL2(R) * Representations of classical Lie groups


Notes


References

*


External links

*{{springer, title=General linear group, id=p/g043680
"GL(2, ''p'') and GL(3, 3) Acting on Points"
by Ed Pegg, Jr., Wolfram Demonstrations Project, 2007. Abstract algebra Linear algebra Lie groups Linear algebraic groups