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).
Definition and basic examples A group G is said to be linear if there exists a field K, an integer d and an injective homomorphism from G to the general linear group GLd (K) (a faithful linear representation of dimension d over K): if needed one can mention the field and dimension by saying that G is linear of degree d over K. Basic instances are groups which are defined as subgroups of a linear group, for example:
The group GLn(K) itself; The special linear group SLn(K) (the subgroup of matrices with determinant 1); The group of invertible upper (or lower) triangular matrices If gi is a collection of elements in GLn(K) indexed by a set I, then the subgroup generated by the gi is a linear group. In the study of Lie groups, it is sometimes pedagogically convenient to restrict attention to Lie groups that can be faithfully represented over the field of complex numbers. (Some authors require that the group be represented as a closed subgroup of the GLn(C).) Books that follow this approach include Hall (2015) and Rossmann (2002).
Classes of linear groups
Classical groups and related examples 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.
Finite 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.
Finitely generated matrix 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 (
2 1 1 0 ) , (
0 1 1 2 ) {\displaystyle {\bigl (}{}_{2}^{1}\,_{1}^{0}{\bigr )},\,{\bigl (}{}_{0}^{1}\,_{1}^{2}{\bigr )}} 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. The mapping class group of a genus 2 surface is also known to be linear.
Examples from geometry 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.
Properties While linear groups are a vast class of examples, among all infinite groups they are distinguished by many remarkable properties. Finitely generated linear groups have the following properties:
They are residually finite; Burnside's theorem: a torsion group of finite exponent which is linear over a field of characteristic 0 must be finite; Schur's theorem: a torsion linear group is locally finite. In particular, if it is finitely generated then it is finite. Selberg's lemma: any finitely generated linear group contains a torsion-free subgroup of finite index. The Tits alternative states that a linear group either contains a non-abelian free group or else is virtually solvable (that is, contains a solvable group of finite index). This has many further consequences, for example:
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