Матрицалық топтар және сызықтық өкілдіктер

In mathematics, a matrix group is a group G consisting of invertible matrices over a specified field K, with the operation of matrix multiplication. A linear group is a group that is isomorphic to a matrix group (that is, 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 groups which are "too big" (for example, the group of permutations of an infinite set), or which exhibit some pathological behavior (for example, finitely generated infinite torsion groups).

The so called classical groups generalize the examples 1 and 2 above. They arise as linear algebraic groups, that is, as subgroups of GLn defined by a finite number of equations. Basic examples are orthogonal, unitary and symplectic groups but it is possible to construct more using division algebras (for example the unit group of a quaternion algebra is a classical group). Note that the projective groups associated to these groups are also linear, though less obviously. For example, the group PSL2(R) is not a group of 2 × 2 matrices, but it has a faithful representation as 3 × 3 matrices (the adjoint representation), which can be used in the general case. Many Lie groups are linear, but not all of them. The universal cover of SL2(R) is not linear, as are many solvable groups, for instance the quotient of the Heisenberg group by a central cyclic subgroup. Discrete subgroups of classical Lie groups (for example lattices or thin groups) are also examples of interesting linear groups.

A finite group G of order n is linear of degree at most n over any field K. This statement is sometimes called Cayley's theorem, and simply results from the fact that the action of G on the group ring K[G] by left (or right) multiplication is linear and faithful. The finite groups of Lie type (classical groups over finite fields) are an important family of finite simple groups, as they take up most of the slots in the classification of finite simple groups.

While example 4 above is too general to define a distinctive class (it includes all linear groups), restricting to a finite index set I, that is, to finitely generated groups allows to construct many interesting examples. For example:
The ping pong lemma can be used to construct many examples of linear groups which are free groups (for instance the group generated by is free). Arithmetic groups are known to be finitely generated. On the other hand, it is a difficult problem to find an explicit set of generators for a given arithmetic group. Braid groups (which are defined as a finitely presented group) have faithful linear representation on a finite dimensional complex vector space where the generators act by explicit matrices.

In some cases the fundamental group of a manifold can be shown to be linear by using representations coming from a geometric structure. For example, all closed surfaces of genus at least 2 are hyperbolic Riemann surfaces. Via the uniformization theorem this gives rise to a representation of its fundamental group in the isometry group of the hyperbolic plane, which is isomorphic to PSL2(R) and this realizes the fundamental group as a Fuchsian group. A generalization of this construction is given by the notion of a (G,X) structure on a manifold. Another example is the fundamental group of Seifert manifolds. On the other hand, it is not known whether all fundamental groups of 3–manifolds are linear.

It is not hard to give infinitely generated examples of non linear groups: for example the infinite abelian group (Z/2Z)N x (Z/3Z)N cannot be linear. Since the symmetric group on an infinite set contains this group it is also not linear. Finding finitely generated examples is subtler and usually requires the use of one of the properties listed above. Since any finitely linear group is residually finite, it cannot be both simple and infinite. Thus finitely generated infinite simple groups, for example Thompson's group F, and the quotient of Higman's group by a maximal proper normal subgroup, are not linear. By the corollary to the Tits alternative mentioned above, groups of intermediate growth such as Grigorchuk's group are not linear. Again by the Tits alternative, as mentioned above all counterexamples to the von Neumann conjecture are not linear. This includes Thompson's group F and Tarski monster groups. By Burnside's theorem, infinite, finitely generated torsion groups such as Tarski monster groups cannot be linear. There are examples of hyperbolic groups which are not linear, obtained as quotients of lattices in the Lie groups Sp(n, 1). The outer automorphism group Out(Fn) of the free group is known not to be linear for n at least 4. In contrast with the case of braid groups, it is an open question whether the mapping class group of a surface of genus > 1 is linear.