TheInfoList

OR:

In
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 ...
, 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. This forms a group, 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 ar ...
, hence the vectors/points they define are in general linear position, 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 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 re ...
) 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 ''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 form ...
s), or a ring ''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 language o ...
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 gr ...
, not necessarily written as matrices. The special linear group, written or SL''n''(''F''), is the 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 an ...
of 1. The group and its subgroups 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 representations, and also arise in the study of spatial symmetries 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 ...
s in general, as well as the study of
polynomials 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 exampl ...
. 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 ...
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 ...
over the field ''F'', the general linear group of ''V'', written GL(''V'') or Aut(''V''), is the group of all
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 automorphism ...
s 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 coordi ...
''n'', then GL(''V'') 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 i ...
. 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 of ...
in ''V''. Given a basis of ''V'' and an automorphism ''T'' in GL(''V''), we have then for every basis vector ''e''''i'' that : $T\left(e_i\right) = \sum_^n a_ e_j$ 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'' ''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 ...
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 an ...
is nonzero. Therefore, an alternative definition of is as the group of matrices with nonzero determinant. Over a
commutative ring In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
''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'' (al ...
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 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 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 additi ...
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 an ...
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 exam ...
map, and hence is an open affine subvariety of M''n''(R) (a non-empty open subset of M''n''(R) in the Zariski topology), 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 In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
of , denoted $\mathfrak_n,$ consists of all real matrices with the commutator 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, 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 compon ...
, 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, 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 In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, ...
of is isomorphic to that of SO(''n''). The homeomorphism also shows that the group is noncompact. “The” maximal compact subgroup of is the
orthogonal group In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. ...
O(''n''), while "the" maximal compact subgroup of is the
special orthogonal group In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. ...
SO(''n''). As for SO(''n''), the group is not simply connected (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 classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, ...
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 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 form ...
s, , is a ''complex''
Lie group 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 additi ...
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 — concretely, that commute with a matrix ''J'' such that , where ''J'' corresponds to multiplying by the imaginary unit ''i''. The
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
corresponding to consists of all complex matrices with the commutator 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 subgroup 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 In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, ...
isomorphic to Z.

# Over finite fields 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, subt ...
with ''q'' elements, then we sometimes write instead of . When ''p'' is prime, is the outer automorphism group of the group Z, and also the
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 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: : $\prod_^\left(q^n-q^k\right)=\left(q^n - 1\right)\left(q^n - q\right)\left(q^n - q^2\right)\ \cdots\ \left(q^n - q^\right).$ 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, the linear span (also called the linear hull or just span) of a set of vectors (from a vector space), denoted , pp. 29-30, §§ 2.5, 2.8 is defined as the set of all linear combinations of the vectors in . It can be characterized ...
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 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 g ...
. 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 Nth root, ...
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 an ...
1. They are special in that they lie on a
subvariety A subvariety (Latin: ''subvarietas'') in botanical nomenclature is a taxonomic rank. They are rarely used to classify organisms. Plant taxonomy Subvariety is ranked: *below that of variety (''varietas'') *above that of form (''forma''). Subvar ...
– 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, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G ...
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 referred ...
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 learni ...
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 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 i ...
to ''F''×. In fact, can be written as a
semidirect product In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. There are two closely related concepts of semidirect product: * an ''inner'' semidirect product is a particular way in wh ...
: :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 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 $n \ne 2$ 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, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
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). The de ...
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 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\times n square matrix with ones on the main diagonal and zeros elsewhere. Terminology and notation The identity matrix is often denoted by I_n, or simply by I if the size is immaterial or ...
. The set of all nonzero scalar matrices forms a subgroup of isomorphic to ''F''×. This group is the center 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 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 are subgroups of GL(''V'') which preserve some sort of
bilinear form In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called '' scalars''). In other words, a bilinear form is a function that is linear ...
on a vector space ''V''. These include the *
orthogonal group In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. ...
, O(''V''), which preserves a non-degenerate
quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to ...
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 gr ...
, 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), *
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 allows ...
on ''V''. These groups provide important examples of Lie groups.

# Related groups and monoids

## Projective linear group

The projective linear group and the projective special linear group are the quotients of and by their centers (which consist of the multiples of the identity matrix therein); they are the induced action 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 ...
.

## 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. Re ...
is an extension of by the group of translations in ''F''''n''. It can be written as a
semidirect product In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. There are two closely related concepts of semidirect product: * an ''inner'' semidirect product is a particular way in wh ...
: :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 transformation In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles. More generally, ...
s 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 phy ...
, .

## 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 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, 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 id ...
), 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 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 ...
, 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, pro ...
.

## 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. Monoids ar ...
, 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 A stable is a building in which livestock, especially horses, are kept. It most commonly means a building that is divided into separate stalls for individual animals and livestock. There are many different types of stables in use today; the ...
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 o ...
to define 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 In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally ...
, which is a larger group, and topologically much simpler, namely contractible – see
Kuiper's theorem In mathematics, Kuiper's theorem (after Nicolaas Kuiper) is a result on the topology of operators on an infinite-dimensional, complex Hilbert space ''H''. It states that the space GL(''H'') of invertible bounded endomorphisms of ''H'' is suc ...
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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 al ...

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