Matrix group

In mathematics, a matrix group is a group G consisting of invertible matrices over some field K, usually fixed in advance, with operations of matrix multiplication and inversion. More generally, one can consider n × n matrices over a commutative ring 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 faithful, finite-dimensional representation over K.

Any finite group is linear, because it can be realized by permutation matrices using Cayley's theorem. 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 M_R(n,n) of n × n matrices over a commutative ring R is itself a ring under matrix addition and multiplication. The group of units of M_R(n,n) is called the general linear group of n × n matrices over the ring R and is denoted GL_n(R) or GL(n,R). All matrix groups are subgroups of some general linear group.

Classical groups

Main article: Classical group

Some particularly interesting matrix groups are the so-called classical groups. 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 groups.

Finite groups as matrix groups

Every finite group is isomorphic to some matrix group. This is similar to Cayley's theorem which states that every finite group is isomorphic to some permutation group. 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,...,g_k} be a generating set for G. The general linear group GL_n(C) of n×n matrices over the complex numbers acts naturally on the vector space Cⁿ. Let B={b₁,…,b_n} be the standard basis for Cⁿ. For each g_i let M_i in GL_n(C) be the matrix which sends each b_j to b_gi(j). That is, if the permutation g_i sends the point j to k then M_i sends the basis vector b_j to b_k. Let M be the subgroup of GL_n(C) generated by {M₁,…,M_k}. 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 g_i to M_i 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 group on 3 points. Let g₁ = (1,2,3) and g₂ = (1,2). Then

$M_1 = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{bmatrix}$

$M_2 = \begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix}$

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 theory studies homomorphisms from a group into a matrix group and character theory studies homomorphisms from a group into a field given by the trace of a representation.

Examples

• See table of Lie groups, 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 groups B_n are linear for all n.^[1]

References

• Brian C. Hall Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, 1st edition, Springer, 2006. ISBN 0-387-40122-9
• Wulf Rossmann, Lie Groups: An Introduction Through Linear Groups (Oxford Graduate Texts in Mathematics), Oxford University Press ISBN 0-19-859683-9.
• La géométrie des groupes classiques, J. Dieudonné. Springer, 1955. ISBN 1-114-75188-X
• The classical groups, H. Weyl, ISBN 0-691-05756-7

1. ^ Stephen J. Bigelow (December 13, 2000), "Braid groups are linear", Journal of the American Mathematical Society 14 (2): 471–486, http://www.ams.org/jams/2001-14-02/S0894-0347-00-00361-1/S0894-0347-00-00361-1.pdf 

External links

• Linear groups, Encyclopaedia of Mathematics