In
mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a
matrix group is a
groupIn mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
G consisting of
invertible matrices over some
fieldIn abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
K, usually fixed in advance, with operations of
matrix multiplicationIn mathematics, matrix multiplication is a binary operation that takes a pair of matrices, and produces another matrix. If A is an n-by-m matrix and B is an m-by-p matrix, the result AB of their multiplication is an n-by-p matrix defined only if the number of columns m of the left matrix A is the...
and inversion. More generally, one can consider
n ×
n matrices over a commutative
ringIn mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
R. (The size of the matrices is restricted to be finite, as any group can be represented as a group of infinite matrices over any field.) A
linear group is an abstract group that is isomorphic to a matrix group over a field
K, in other words, admitting a
faithfulIn mathematics, especially in the area of abstract algebra known as representation theory, a faithful representation ρ of a group G on a vector space V is a linear representation in which different elements g of G are represented by distinct linear mappings ρ.In more abstract language, this means...
, finite-dimensional
representationIn the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication...
over
K.
Any
finite groupIn mathematics and abstract algebra, a finite group is a group whose underlying set G has finitely many elements. During the twentieth century, mathematicians investigated certain aspects of the theory of finite groups in great depth, especially the local theory of finite groups, and the theory of...
is linear, because it can be realized by permutation matrices using
Cayley's theoremIn group theory, Cayley's theorem, named in honor of Arthur Cayley, states that every group G is isomorphic to a subgroup of the symmetric group acting on G...
. Among infinite groups, linear groups form an interesting and tractable class. Examples of groups that are not linear include all "sufficiently large" groups; for example, the infinite symmetric group of permutations of an infinite set.
Basic examples
The set
MR(
n,
n) of
n ×
n matrices over a
commutative ringIn ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
R is itself a ring under matrix addition and multiplication. The group of units of
MR(
n,
n) is called the
general linear groupIn mathematics, the general linear group of degree n is the set of n×n invertible matrices, 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...
of
n ×
n matrices over the ring
R and is denoted
GLn(
R) or
GL(
n,
R). All matrix groups are subgroups of some general linear group.
Classical groups
Some particularly interesting matrix groups are the so-called
classical groupIn mathematics, the classical Lie groups are four infinite families of Lie groups closely related to the symmetries of Euclidean spaces. Their finite analogues are the classical groups of Lie type...
s. When the ring of coefficients of the matrix group is the real numbers, these groups are the classical Lie groups. When the underlying ring is a finite field the classical groups are groups of Lie type. These groups play an important role in the
classification of finite simple groupsIn mathematics, the classification of the finite simple groups is a theorem stating that every finite simple group belongs to one of four categories described below. These groups can be seen as the basic building blocks of all finite groups, in much the same way as the prime numbers are the basic...
.
Finite groups as matrix groups
Every finite group is isomorphic to some matrix group. This is similar to
Cayley's theoremIn group theory, Cayley's theorem, named in honor of Arthur Cayley, states that every group G is isomorphic to a subgroup of the symmetric group acting on G...
which states that every finite group is isomorphic to some
permutation groupIn mathematics, a permutation group is a group G whose elements are permutations of a given set M, and whose group operation is the composition of permutations in G ; the relationship is often written as...
. Since the isomorphism property is transitive one need only consider how to form a matrix group from a permutation group.
Let
G be a permutation group on
n points (Ω = {1,2,…,n}) and let {
g1,...,
gk} be a generating set for
G. The general linear group
GLn(
C) of
n×
n matrices over the complex numbers acts naturally on the vector space
Cn. Let
B={
b1,…,
bn} be the standard basis for
Cn. For each
gi let
Mi in
GLn(
C) be the matrix which sends each
bj to
bgi(j). That is, if the permutation
gi sends the point
j to
k then
Mi sends the basis vector
bj to
bk. Let
M be the subgroup of
GLn(
C) generated by {
M1,…,
Mk}. The action of
G on Ω is then precisely the same as the action of
M on
B. It can be proved that the function taking each
gi to
Mi extends to an isomorphism and thus every group is isomorphic to a matrix group.
Note that the field (
C in the above case) is irrelevant since
M contains only elements with entries 0 or 1. One can just as easily perform the construction for an arbitrary field since the elements 0 and 1 exist in every field.
As an example, let
G =
S3, the
symmetric groupIn mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...
on 3 points. Let
g1 = (1,2,3) and
g2 = (1,2). Then
-
-
Notice that M1b1 = b2, M1b2 = b3 and M1b3 = b1. Likewise, M2b1 = b2, M2b2 = b1 and M2b3 = b3.
Representation theory and character theory
Linear transformations and matrices are (generally speaking) well-understood objects in mathematics and have been used extensively in the study of groups. In particular representation theoryIn the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication...
studies homomorphisms from a group into a matrix group and character theoryIn mathematics, more specifically in group theory, the character of a group representation is a function on the group which associates to each group element the trace of the corresponding matrix....
studies homomorphisms from a group into a field given by the trace of a representation.
Examples
- See table of Lie groups
This article gives a table of some common Lie groups and their associated Lie algebras.The following are noted: the topological properties of the group , as well as on their algebraic properties .For more examples of Lie groups and other...
, list of finite simple groups, and list of simple Lie groups for many examples.
- See list of transitive finite linear groups.
- In 2000 a longstanding conjecture was resolved when it was shown that the braid group
In mathematics, the braid group on n strands, denoted by Bn, is a group which has an intuitive geometrical representation, and in a sense generalizes the symmetric group Sn. Here, n is a natural number; if n > 1, then Bn is an infinite group...
s Bn are linear for all n.
External links