In
mathematics, a matrix group is 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 ...
''G'' consisting of
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 ...
matrices
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
over a specified
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 ...
''K'', with the operation of
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 ...
. A linear group is a group that is
isomorphic to a matrix group (that is, admitting a
faithful, finite-dimensional
representation over ''K'').
Any
finite group is linear, because it can be realized by
permutation matrices
In mathematics, particularly in matrix theory, a permutation matrix is a square binary matrix that has exactly one entry of 1 in each row and each column and 0s elsewhere. Each such matrix, say , represents a permutation of elements and, when ...
using
Cayley's theorem
In group theory, Cayley's theorem, named in honour of Arthur Cayley, states that every group is isomorphic to a subgroup of a symmetric group.
More specifically, is isomorphic to a subgroup of the symmetric group \operatorname(G) whose elem ...
. Among
infinite groups, linear groups form an interesting and tractable class. Examples of groups that are not linear include groups which are "too big" (for example, the group of permutations of an infinite set), or which exhibit some pathological behavior (for example,
finitely generated infinite
torsion group
In group theory, a branch of mathematics, a torsion group or a periodic group is a group in which every element has finite order. The exponent of such a group, if it exists, is the least common multiple of the orders of the elements.
For examp ...
s).
Definition and basic examples
A group ''G'' is said to be ''linear'' if there exists a field ''K'', an
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 ...
''d'' and an
injective homomorphism
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...
from ''G'' to the
general linear group
In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
GL
''d''(''K'') (a faithful linear
representation of dimension ''d'' over ''K''): if needed one can mention the field and dimension by saying that ''G'' is ''linear of degree d over K''. Basic instances are groups which are defined as
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 of a linear group, for example:
#The group GL
''n''(''K'') itself;
#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 ...
SL
''n''(''K'') (the subgroup of 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);
#The group of invertible upper (or lower)
triangular matrices
#If ''g
i'' is a collection of elements in GL
''n''(''K'')
indexed by a set ''I'', then the subgroup generated by the ''g
i'' is a linear group.
In the study of
Lie groups, it is sometimes pedagogically convenient to restrict attention to Lie groups that can be faithfully represented 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. (Some authors require that the group be represented as a ''closed'' subgroup of the GL
''n''(C).) Books that follow this approach include Hall (2015) and Rossmann (2002).
Classes of linear groups
Classical groups and related examples
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 generalize the examples 1 and 2 above. They arise as
linear algebraic group
In mathematics, a linear algebraic group is a subgroup of the group of invertible n\times n matrices (under matrix multiplication) that is defined by polynomial equations. An example is the orthogonal group, defined by the relation M^TM = I_n ...
s, that is, as subgroups of GL
''n'' defined by a finite number of equations. Basic examples are
orthogonal,
unitary
Unitary may refer to:
Mathematics
* Unitary divisor
* Unitary element
* Unitary group
* Unitary matrix
* Unitary morphism
* Unitary operator
* Unitary transformation
* Unitary representation
* Unitarity (physics)
* ''E''-unitary inverse semigrou ...
and
symplectic groups but it is possible to construct more using
division algebra
In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division, except by zero, is always possible.
Definitions
Formally, we start with a non-zero algebra ''D'' over a fie ...
s (for example 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 a
quaternion algebra
In mathematics, a quaternion algebra over a field ''F'' is a central simple algebra ''A'' over ''F''See Milies & Sehgal, An introduction to group rings, exercise 17, chapter 2. that has dimension 4 over ''F''. Every quaternion algebra becomes a ma ...
is a classical group). Note that the
projective groups associated to these groups are also linear, though less obviously. For example, the group PSL
2(R) is not a group of 2 × 2 matrices, but it has a faithful representation as 3 × 3 matrices (the
adjoint representation
In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is G ...
), which can be used in the general case.
Many
Lie groups are linear, but not all of them. The
universal cover of SL2(R) is not linear, as are many
solvable group
In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions. Equivalently, a solvable group is a group whose derived series terminate ...
s, for instance the
quotient
In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...
of the
Heisenberg group
In mathematics, the Heisenberg group H, named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form
::\begin
1 & a & c\\
0 & 1 & b\\
0 & 0 & 1\\
\end
under the operation of matrix multiplication. Elements ...
by a
central cyclic subgroup.
Discrete subgroup
In mathematics, a topological group ''G'' is called a discrete group if there is no limit point in it (i.e., for each element in ''G'', there is a neighborhood which only contains that element). Equivalently, the group ''G'' is discrete if and on ...
s of classical Lie groups (for example
lattices or
thin groups) are also examples of interesting linear groups.
Finite groups
A finite group ''G'' of
order ''n'' is linear of degree at most ''n'' over any field ''K''. This statement is sometimes called Cayley's theorem, and simply results from the fact that the action of ''G'' on the
group ring
In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the giv ...
''K''
'G''by left (or right) multiplication is linear and faithful. The
finite groups of Lie type (classical groups over finite fields) are an important family of finite
simple group
SIMPLE Group Limited is a conglomeration of separately run companies that each has its core area in International Consulting. The core business areas are Legal Services, Fiduciary Activities, Banking Intermediation and Corporate Service.
The d ...
s, as they take up most of the slots in the
classification of finite simple groups
In mathematics, the classification of the finite simple groups is a result of group theory stating that every finite simple group is either cyclic, or alternating, or it belongs to a broad infinite class called the groups of Lie type, or else i ...
.
Finitely generated matrix groups
While example 4 above is too general to define a distinctive class (it includes all linear groups), restricting to a finite index set ''I'', that is, to
finitely generated group
In algebra, a finitely generated group is a group ''G'' that has some finite generating set ''S'' so that every element of ''G'' can be written as the combination (under the group operation) of finitely many elements of ''S'' and of inverses o ...
s allows to construct many interesting examples. For example:
*The
ping-pong lemma can be used to construct many examples of linear groups which are
free group
In mathematics, the free group ''F'S'' over a given set ''S'' consists of all words that can be built from members of ''S'', considering two words to be different unless their equality follows from the group axioms (e.g. ''st'' = ''suu''−1' ...
s (for instance the group generated by
is free).
*
Arithmetic group
In mathematics, an arithmetic group is a group obtained as the integer points of an algebraic group, for example \mathrm_2(\Z). They arise naturally in the study of arithmetic properties of quadratic forms and other classical topics in number the ...
s are known to be finitely generated. On the other hand, it is a difficult problem to find an explicit set of generators for a given arithmetic group.
*
Braid group
A braid (also referred to as a plait) is a complex structure or pattern formed by interlacing two or more strands of flexible material such as textile yarns, wire, or hair.
The simplest and most common version is a flat, solid, three-strande ...
s (which are defined as a
finitely presented group
In mathematics, a presentation is one method of specifying a group. A presentation of a group ''G'' comprises a set ''S'' of generators—so that every element of the group can be written as a product of powers of some of these generators—and ...
) have faithful linear representation on a
finite-dimensional complex vector space where the generators act by explicit matrices.
Examples from geometry
In some cases the
fundamental group of a
manifold can be shown to be linear by using representations coming from a geometric structure. For example, all
closed surface
In the part of mathematics referred to as topology, a surface is a two-dimensional manifold. Some surfaces arise as the boundaries of three-dimensional solids; for example, the sphere is the boundary of the solid ball. Other surfaces arise as g ...
s of
genus
Genus ( plural genera ) is a taxonomic rank used in the biological classification of living and fossil organisms as well as viruses. In the hierarchy of biological classification, genus comes above species and below family. In binomial nom ...
at least 2 are hyperbolic
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ver ...
s. Via the
uniformization theorem
In mathematics, the uniformization theorem says that every simply connected Riemann surface is conformally equivalent to one of three Riemann surfaces: the open unit disk, the complex plane, or the Riemann sphere. The theorem is a generalization ...
this gives rise to a representation of its fundamental group in the
isometry group In mathematics, the isometry group of a metric space is the set of all bijective isometries (i.e. bijective, distance-preserving maps) from the metric space onto itself, with the function composition as group operation. Its identity element is the ...
of the
hyperbolic plane
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For any given line ''R'' and point ''P' ...
, which is isomorphic to PSL
2(R) and this realizes the fundamental group as a
Fuchsian group
In mathematics, a Fuchsian group is a discrete subgroup of PSL(2,R). The group PSL(2,R) can be regarded equivalently as a group of isometries of the hyperbolic plane, or conformal transformations of the unit disc, or conformal transformations o ...
. A generalization of this construction is given by the notion of a
(''G'',''X'')-structure on a manifold.
Another example is the fundamental group of
Seifert manifold
A Seifert fiber space is a 3-manifold together with a decomposition as a disjoint union of circles. In other words, it is a S^1-bundle (circle bundle) over a 2-dimensional orbifold. Many 3-manifolds are Seifert fiber spaces, and they account for ...
s. On the other hand, it is not known whether all fundamental groups of 3–manifolds are linear.
Properties
While linear groups are a vast class of examples, among all infinite groups they are distinguished by many remarkable properties. Finitely generated linear groups have the following properties:
*They are
residually finite {{unsourced, date=September 2022
In the mathematical field of group theory, a group ''G'' is residually finite or finitely approximable if for every element ''g'' that is not the identity in ''G'' there is a homomorphism ''h'' from ''G'' to a fini ...
;
*
Burnside's theorem: a
torsion
Torsion may refer to:
Science
* Torsion (mechanics), the twisting of an object due to an applied torque
* Torsion of spacetime, the field used in Einstein–Cartan theory and
** Alternatives to general relativity
* Torsion angle, in chemistry
Bi ...
group of finite
exponent which is linear over a field of characteristic 0 must be finite;
*Schur's theorem: a
torsion
Torsion may refer to:
Science
* Torsion (mechanics), the twisting of an object due to an applied torque
* Torsion of spacetime, the field used in Einstein–Cartan theory and
** Alternatives to general relativity
* Torsion angle, in chemistry
Bi ...
linear group is
locally finite. In particular, if it is finitely generated then it is finite.
*Selberg's lemma: any finitely generated linear group contains a
torsion-free subgroup of finite
index.
The
Tits alternative In mathematics, the Tits alternative, named for Jacques Tits, is an important theorem about the structure of finitely generated linear groups.
Statement
The theorem, proven by Tits, is stated as follows.
Consequences
A linear group is not a ...
states that a linear group either contains a non-abelian free group or else is
virtually
In mathematics, especially in the area of abstract algebra that studies infinite groups, the adverb virtually is used to modify a property so that it need only hold for a subgroup of finite index. Given a property P, the group ''G'' is said to b ...
solvable (that is, contains a
solvable group
In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions. Equivalently, a solvable group is a group whose derived series terminate ...
of finite index). This has many further consequences, for example:
*the
Dehn function
In the mathematical subject of geometric group theory, a Dehn function, named after Max Dehn, is an optimal function associated to a finite group presentation which bounds the ''area'' of a ''relation'' in that group (that is a freely reduced word ...
of a finitely generated linear group can only be either polynomial or exponential;
*an
amenable linear group is virtually solvable, in particular
elementary amenable In mathematics, a group is called elementary amenable if it can be built up from finite groups and abelian groups by a sequence of simple operations that result in amenable groups when applied to amenable groups. Since finite groups and abelian gro ...
;
*the
von Neumann conjecture In mathematics, the von Neumann conjecture stated that a group (mathematics), group ''G'' is non-Amenable group, amenable if and only if ''G'' contains a subgroup that is a free group on two Generating set of a group, generators. The conjecture was ...
is true for linear groups.
Examples of non-linear groups
It is not hard to give infinitely generated examples of non-linear groups: for example the infinite abelian group (Z/2Z)
N x (Z/3Z)
N cannot be linear. Since 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 ...
on an infinite set contains this group it is also not linear. Finding finitely generated examples is subtler and usually requires the use of one of the properties listed above.
*Since any finitely linear group is residually finite, it cannot be both simple and infinite. Thus finitely generated infinite simple groups, for example
Thompson's group ''F'', and
Higman's group, are not linear.
*By the corollary to the Tits alternative mentioned above, groups of intermediate growth such as
Grigorchuk's group are not linear.
*Again by the Tits alternative, as mentioned above all counterexamples to the
von Neumann conjecture In mathematics, the von Neumann conjecture stated that a group (mathematics), group ''G'' is non-Amenable group, amenable if and only if ''G'' contains a subgroup that is a free group on two Generating set of a group, generators. The conjecture was ...
are not linear. This includes
Thompson's group ''F'' and
Tarski monster groups.
*By Burnside's theorem, infinite, finitely generated torsion groups such as
Tarski monster groups cannot be linear.
*There are examples of
hyperbolic group
In group theory, more precisely in geometric group theory, a hyperbolic group, also known as a ''word hyperbolic group'' or ''Gromov hyperbolic group'', is a finitely generated group equipped with a word metric satisfying certain properties abstra ...
s which are not linear, obtained as quotients of lattices in the Lie groups Sp(''n'', 1).
*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 ...
Out(F''n'') of the free group is known not to be linear for ''n'' at least 4.
*In contrast with the case of braid groups, it is an
open question whether the
mapping class group of a surface of genus > 1 is linear.
Representation theory
Once a group has been established to be linear it is interesting to try to find "optimal" faithful linear representations for it, for example of the lowest possible dimension, or even to try to classify all its linear representations (including those which are not faithful). These questions are the object of
representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...
. Salient parts of the theory include:
*
Representation theory of finite groups
The representation theory of groups is a part of mathematics which examines how groups act on given structures.
Here the focus is in particular on operations of groups on vector spaces. Nevertheless, groups acting on other groups or on sets are ...
;
*
Representation theory of Lie groups
In mathematics and theoretical physics, a representation of a Lie group is a linear action of a Lie group on a vector space. Equivalently, a representation is a smooth homomorphism of the group into the group of invertible operators on the vec ...
and more generally linear algebraic groups.
The representation theory of infinite finitely generated groups is in general mysterious; the object of interest in this case are the
character varieties of the group, which are well understood only in very few cases, for example free groups, surface groups and more generally lattices in Lie groups (for example through Margulis'
superrigidity theorem and other rigidity results).
Notes
References
* .
* .
*
*{{cite book , last=Wehrfritz , first=B.A.F. , title=Infinite linear groups , publisher=Springer-Verlag , series=Ergebnisse der Mathematik und ihrer Grenzgebiete , volume=76 , year=1973
Infinite group theory
Matrices