# 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

{{#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 {*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

*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) 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*are linear for all_{n}*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

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## Further reading

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## External links

*Linear groups*, Encyclopaedia of Mathematics