 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, finitedimensional 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.
Contents
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 groupSome particularly interesting matrix groups are the socalled 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 {g_{1},...,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 C^{n}. Let B={b_{1},…,b_{n}} be the standard basis for C^{n}. 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 {M_{1},…,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 = S_{3}, the symmetric group on 3 points. Let g_{1} = (1,2,3) and g_{2} = (1,2). Then
Notice that M_{1}b_{1} = b_{2}, M_{1}b_{2} = b_{3} and M_{1}b_{3} = b_{1}. Likewise, M_{2}b_{1} = b_{2}, M_{2}b_{2} = b_{1} and M_{2}b_{3} = b_{3}.
Representation theory and character theory
Linear transformations and matrices are (generally speaking) wellunderstood 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 0387401229
 Wulf Rossmann, Lie Groups: An Introduction Through Linear Groups (Oxford Graduate Texts in Mathematics), Oxford University Press ISBN 0198596839.
 La géométrie des groupes classiques, J. Dieudonné. Springer, 1955. ISBN 111475188X
 The classical groups, H. Weyl, ISBN 0691057567
 ^ 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/20011402/S0894034700003611/S0894034700003611.pdf
External links
 Linear groups, Encyclopaedia of Mathematics
Categories: Infinite group theory
 Matrices
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