- Linear group
In
mathematics , a matrix group is a group "G" consisting of invertible matrices over some field "K", usually fixed in advance, with operations ofmatrix 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 bypermutation matrices usingCayley'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. Thegroup of units of "M""R"("n","n") is called thegeneral 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
Some particularly interesting matrix groups are the so-called
classical group s. When the ring of coefficients of the matrix group is the real numbers, these groups are theclassical Lie group s. When the underlying ring is a finite field the classical groups aregroups of Lie type . These groups play an important role in theclassification 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 somepermutation 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""g""i"("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 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) 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 , andlist of simple Lie groups for many examples.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 0-691-05756-7External links
* [http://eom.springer.de/L/l059250.htm EoM article "Linear groups"]
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