Matrix group

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In mathematics, a matrix group is a group where each element is a matrix, and the binary operation is given by matrix multiplication.

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.

1 Classical groups
2 Finite groups as matrix groups
3 Representation theory and character theory
4 Examples
5 References

[edit] Classical groups

Some particularly interesting matrix groups are the so-called classical groups. When the underlying ring of the matrix group is the real numbers, these matrix groups define Lie groups. When the underlying ring is a finite field the matrix groups define groups of Lie type. These groups play an important role in the classification of finite simple groups. Many linear groups have a "special" subgroup, usually consisting of the elements of determinant 1 (though for orthogonal groups in characteristic 2 it consists of the elements of Dickson invariant 0), and most of them have associated "projective" quotients, which are the quotients by the center of the group. The word "general" in front of a group name usually means that the group is allowed to multiply some sort of form by a constant, rather than leaving it fixed.

The general linear group GL_n(R) is the group of all automorphisms of some module. There is a subgroup the special linear group SL_n(R), and their quotients the projective general linear group PGL_n(R) = GL_n(R)/Z(GL_n(R)) and the projective special linear group PSL_n(R) = SL_n(R)/Z(SL_n(R)). The projective special linear group PSL_n(R) over a field R is simple for n≥2, except for the 2 cases when n=2 and the field has order 2 or 3.
The unitary group U_n(R) is a group preseving a sesquilinear form on a module. There is a subgroup, the special unitary group SU_n(R) and their quotients the projective unitary group PU_n(R) = U_n(R)/Z(U_n(R)) and the projective special unitary group PSU_n(R) = SU_n(R)/Z(SU_n(R))
The symplectic group Sp_2n(R) preserves a skew symmetric form on a module. It has a quotient, the projective symplectic group PSp_2n(R). The general symplectic group GSp_2n(R) consists of the automorphisms of a module multiplying a skew symmetric form by some invertible scalar. The projective symplectic group PSp_2n(R) over a field R is simple for n≥1, except for the 2 cases when n=1 and the field has order 2 or 3.
The orthogonal group O_n(R) preserves a non-degenerate quadratic form on a module. There is a subgroup, the special orthogonal group SO_n(R) and their quotients the projective orthogonal group PO_n(R), and the projective special orthogonal group PSO_n(R). (In characteristic 2 the determinant is always 1, so the special orthogonal group is often defined as the subgroup of elements of Dickson invariant 1.) There is a nameless group often denoted by Ω_n(R) consisting of the elements of the orthogonal group of elements of spinor norm 1, with corresponding subgroup and quotient groups SΩ_n(R), PΩ_n(R), PSΩ_n(R). (For positive definite quadratic forms over the reals, the group Ω happens to be the same as the orthogonal group, but in general it is smaller.) There is also a double cover of Ω_n(R), called the pin group Pin_n(R), and it has a subgroup called the spin group Spin_n(R). The general orthogonal group GO_n(R) consists of the automorphisms of a module multiplying a quadratic form by some invertible scalar.

The subscript n usually indicates the dimension of the module on which the group is acting.

[edit] 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 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.

[edit] 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.