In
mathematics, the general linear group of degree ''n'' is the set of
invertible matrices
In linear algebra, an -by- square matrix is called invertible (also nonsingular or nondegenerate), if there exists an -by- square matrix such that
:\mathbf = \mathbf = \mathbf_n \
where denotes the -by- identity matrix and the multiplicati ...
, together with the operation of ordinary
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 ...
. This forms a
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
, 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 (and also the rows) of an invertible matrix 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 ...
, hence the vectors/points they define are in
general linear position
In algebraic geometry and computational geometry, general position is a notion of genericity for a set of points, or other geometric objects. It means the ''general case'' situation, as opposed to some more special or coincidental cases that are ...
, 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) 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 grass ...
''F'' (such as the
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s), or 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 ...
''R'' (such as the ring of
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
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 consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...
, not necessarily written as matrices.
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 ge ...
, written or SL
''n''(''F''), is the
subgroup
In group theory, a branch of mathematics, given a 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, ''H'' is a subgroup ...
of consisting of matrices with a
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 a ...
of 1.
The group and its
subgroup
In group theory, a branch of mathematics, given a 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, ''H'' is a subgroup ...
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, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used to ...
s, and also arise in the study of spatial
symmetries
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
and symmetries of
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 ...
s in general, as well as the study of
polynomials. The
modular group 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 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 ...
over the field ''F'', the general linear group of ''V'', written GL(''V'') or Aut(''V''), is the group of all
automorphisms of ''V'', i.e. the set of all
bijective
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
linear transformation
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
s , together with functional composition as group operation. If ''V'' has finite
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...
''n'', then GL(''V'') and are
isomorphic. The isomorphism is not canonical; it depends on a choice of
basis
Basis may refer to:
Finance and accounting
* Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
* Basis trading, a trading strategy consisting ...
in ''V''. Given a basis of ''V'' and an automorphism ''T'' in GL(''V''), we have then for every basis vector ''e''
''i'' that
:
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'' ''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
In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers.
Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is ...
if and only if its
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 a ...
is nonzero. Therefore, an alternative definition of is as the group of matrices with nonzero determinant.
Over a
commutative ring ''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'' (a ...
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 algebra, a unit of a ring is an invertible element for the multiplication of the ring. That is, an element of a ring is a unit if there exists in such that
vu = uv = 1,
where is the multiplicative identity; the element is unique for this ...
of the
matrix ring .
As a Lie group
Real case
The general linear group over the field of
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s is a real
Lie group 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:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
of dimension ''n''
2. The subset consists of those matrices whose
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 a ...
is non-zero. The determinant is a
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example ...
map, and hence is an
open affine subvariety of M
''n''(R) (a
non-empty
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 t ...
open subset of M
''n''(R) in the
Zariski topology
In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is n ...
), and therefore
a
smooth manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
of the same dimension.
The
Lie algebra of , denoted
consists of all real matrices with the
commutator serving as the Lie bracket.
As a manifold, is not
connected
Connected may refer to:
Film and television
* ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular''
* '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film
* ''Connected'' (2015 TV ...
but rather has two
connected components: the matrices with positive determinant and the ones with negative determinant. The
identity component
In mathematics, specifically group theory, the identity component of a group ''G'' refers to several closely related notions of the largest connected subgroup of ''G'' containing the identity element.
In point set topology, the identity compo ...
, 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
polar decomposition
In mathematics, the polar decomposition of a square real or complex matrix A is a factorization of the form A = U P, where U is an orthogonal matrix and P is a positive semi-definite symmetric matrix (U is a unitary matrix and P is a positive se ...
, which is unique for invertible matrices, shows that there is a homeomorphism between and the Cartesian product of O(''n'') with the set of positive-definite symmetric matrices. Similarly, it shows that there is a homeomorphism between and the Cartesian product of SO(''n'') with the set of positive-definite symmetric matrices. Because the latter is contractible, the
fundamental group of is isomorphic to that of SO(''n'').
The homeomorphism also shows that the group is
noncompact. “The”
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 is the
orthogonal group O(''n''), while "the" maximal compact subgroup of is the
special orthogonal group SO(''n''). As for SO(''n''), the group is not
simply connected (except when , but rather has a
fundamental group isomorphic to Z for or Z
2 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 extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s, , is a ''complex''
Lie group 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 structure
In mathematics, a complex structure on a real vector space ''V'' is an automorphism of ''V'' that squares to the minus identity, −''I''. Such a structure on ''V'' allows one to define multiplication by complex scalars in a canonical fashion so ...
— concretely, that commute with a matrix ''J'' such that , where ''J'' corresponds to multiplying by the imaginary unit ''i''.
The
Lie algebra corresponding to consists of all complex matrices with the
commutator serving as the Lie bracket.
Unlike the real case, is
connected
Connected may refer to:
Film and television
* ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular''
* '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film
* ''Connected'' (2015 TV ...
. This follows, in part, since the multiplicative group of complex numbers C
∗ is connected. The group manifold is not compact; rather its
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 ...
is the
unitary group
In mathematics, the unitary group of degree ''n'', denoted U(''n''), is the group of unitary matrices, with the group operation of matrix multiplication. The unitary group is a subgroup of the general linear group . Hyperorthogonal group is ...
U(''n''). As for U(''n''), the group manifold is not
simply connected but has a
fundamental group isomorphic to Z.
Over finite fields
![Symmetric group 3; Cayley table; GL(2,2)](https://upload.wikimedia.org/wikipedia/commons/2/27/Symmetric_group_3%3B_Cayley_table%3B_GL%282%2C2%29.svg)
If ''F'' is 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 ...
with ''q'' elements, then we sometimes write instead of . When ''p'' is prime, is the
outer automorphism group In mathematics, the outer automorphism group of a group, , is the quotient, , where is the automorphism group of and ) is the subgroup consisting of inner automorphisms. The outer automorphism group is usually denoted . If is trivial and has a t ...
of the group Z, and also the
automorphism group, because Z is abelian, so the
inner automorphism group
In abstract algebra an inner automorphism is an automorphism of a group, ring, or algebra given by the conjugation action of a fixed element, called the ''conjugating element''. They can be realized via simple operations from within the group it ...
is trivial.
The order of is:
:
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 of the first columns. In
''q''-analog notation, this is
.
For example, has order . It is the automorphism group 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 ...
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 -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 projective ...
over ''F'': in other words the number of subspaces of a given dimension ''k''. This requires only finding the order of the
stabilizer subgroup of one such subspace and dividing into the formula just given, by the
orbit-stabilizer theorem
In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...
.
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 simplici ...
s of complex Grassmannians. This was one of the clues leading to the
Weil conjectures
In mathematics, the Weil conjectures were highly influential proposals by . They led to a successful multi-decade program to prove them, in which many leading researchers developed the framework of modern algebraic geometry and number theory.
Th ...
.
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, the field with one element is a suggestive name for an object that should behave similarly to a finite field with a single element, if such a field could exist. This object is denoted F1, or, in a French–English pun, Fun. The name ...
, one thus interprets 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 ...
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 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 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, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the po ...
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, 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 a ...
1. They are special in that they lie on a
subvariety – 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 of .
If we write ''F''
× for the
multiplicative group
In mathematics and group theory, the term multiplicative group refers to one of the following concepts:
*the group under multiplication of the invertible elements of a field, ring, or other structure for which one of its operations is referre ...
of ''F'' (excluding 0), then the determinant is 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)
w ...
: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 learn ...
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 fo ...
, is
isomorphic to ''F''
×. In fact, can be written as a
semidirect product:
:GL(''n'', ''F'') = SL(''n'', ''F'') ⋊ ''F''
×
The special linear group is also the
derived group
In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group.
The commutator subgroup is important because it is the smallest normal ...
(also known as commutator subgroup) of the GL(''n'', ''F'') (for a field or 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 ...
''F'') provided that
or ''k'' is not the
field with two elements.
[, Theorem II.9.4]
When ''F'' is R or C, is a
Lie subgroup
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the add ...
of of dimension . The
Lie algebra 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), a Dutch progressive rock band
* ''The Trace'' (album)
Other uses in arts and entertainment
* ''Trace'' ...
. The Lie bracket is given by the
commutator.
The special linear group can be characterized as the group of ''
volume
Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). Th ...
and
orientation-preserving
The orientation of a real vector space or simply orientation of a vector space is the arbitrary choice of which ordered bases are "positively" oriented and which are "negatively" oriented. In the three-dimensional Euclidean space, right-handed ...
'' linear transformations of R
''n''.
The group is simply connected, while is not. has the same fundamental group as , that is, Z for and Z
2 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. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagonal m ...
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. 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), 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 . 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, 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 the field ''F''.
Classical groups
The so-called
classical group
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 s ...
s are subgroups of GL(''V'') which preserve some sort of
bilinear form on a vector space ''V''. These include the
*
orthogonal group, 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 defin ...
quadratic form on ''V'',
*
symplectic group
In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted and for positive integer ''n'' and field F (usually C or R). The latter is called the compact symplectic gro ...
, Sp(''V''), which preserves a
symplectic form In mathematics, a symplectic vector space is a vector space ''V'' over a field ''F'' (for example the real numbers R) equipped with a symplectic bilinear form.
A symplectic bilinear form is a mapping that is
; Bilinear: Linear in each argument ...
on ''V'' (a non-degenerate
alternating form
In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is a ...
),
*
unitary group
In mathematics, the unitary group of degree ''n'', denoted U(''n''), is the group of unitary matrices, with the group operation of matrix multiplication. The unitary group is a subgroup of the general linear group . Hyperorthogonal group is ...
, 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 in each of its arguments, but a sesquilinear form allow ...
on ''V''.
These groups provide important examples of Lie groups.
Related groups and monoids
Projective linear group
The
projective linear group
In mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector space ''V'' on the associate ...
and the
projective special linear group
In mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector space ''V'' on the associat ...
are the
quotients of and by their
centers
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 ...
(which consist of the multiples of the identity matrix therein); they are 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 ...
on the associated
projective space.
Affine group
The
affine group
In mathematics, the affine group or general affine group of any affine space over a field is the group of all invertible affine transformations from the space into itself.
It is a Lie group if is the real or complex field or quaternions.
Rela ...
is an
extension of by the group of translations in ''F''
''n''. It can be written as a
semidirect product:
: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
In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related ...
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
The Poincaré group, named after Henri Poincaré (1906), was first defined by Hermann Minkowski (1908) as the group of Minkowski spacetime isometries. It is a ten-dimensional non-abelian Lie group that is of importance as a model in our und ...
is the affine group associated to the
Lorentz group
In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physicis ...
, .
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
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 ...
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, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the po ...
of ''F'' (over its
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 ...
), which acts on by the Galois action on the entries.
The main interest of is that the associated
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' ...
(which contains 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 ...
of
projective space, for , and thus semilinear maps are of interest in
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, ...
.
Full linear monoid
If one removes the restriction of the determinant being non-zero, the resulting algebraic structure is a
monoid
In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0.
Monoid ...
, usually called the full linear monoid,
but occasionally also ''full linear semigroup'',
''general linear monoid''
etc. It is actually a
regular semigroup In mathematics, a regular semigroup is a semigroup ''S'' in which every element is regular, i.e., for each element ''a'' in ''S'' there exists an element ''x'' in ''S'' such that . Regular semigroups are one of the most-studied classes of semigroup ...
.
Infinite general linear group
The infinite general linear group or
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
Algebraic ''K''-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are assigned objects called ''K''-groups. These are groups in the sense ...
to define
K1, and over the reals has a well-understood topology, thanks to
Bott periodicity
In mathematics, the Bott periodicity theorem describes a periodicity in the homotopy groups of classical groups, discovered by , which proved to be of foundational significance for much further research, in particular in K-theory of stable comp ...
.
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
A ''list'' is any set of items in a row. List or lists may also refer to:
People
* List (surname)
Organizations
* List College, an undergraduate division of the Jewish Theological Seminary of America
* SC Germania List, German rugby unio ...
*
SL2(R)
*
Representation theory of SL2(R)
*
Representations of classical Lie groups
In mathematics, the finite-dimensional representations of the complex classical Lie groups
GL(n,\mathbb), SL(n,\mathbb), O(n,\mathbb), SO(n,\mathbb), Sp(2n,\mathbb),
can be constructed using the general representation theory of semisimple Lie a ...
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
The Wolfram Demonstrations Project is an organized, open-source collection of small (or medium-size) interactive programs called Demonstrations, which are meant to visually and interactively represent ideas from a range of fields. It is hos ...
, 2007.
Abstract algebra
Linear algebra
Lie groups
Linear algebraic groups