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 MR(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 MR(n,n) is called the general linear group 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

{{#invoke:main|main}} 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,...,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 group on 3 points. Let g1 = (1,2,3) and g2 = (1,2). Then

$M_{1}={\begin{bmatrix}0&0&1\\1&0&0\\0&1&0\end{bmatrix}}$ $M_{2}={\begin{bmatrix}0&1&0\\1&0&0\\0&0&1\end{bmatrix}}$ 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.