Matrix group

In mathematics, a matrix group is a group G consisting of invertible matrices over a specified field K, with the operation of matrix multiplication, and 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. In mathematics, a matrix group is a group G consisting of invertible matrices over a specified field K, with the operation of matrix multiplication, and 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 groups which are 'too big' (for example, the group of permutations of an infinite set), or which exhibit some pathological behaviour (for example finitely generated infinite torsion groups). A group G is said to be linear if there exists a field K, an integer d and an injective morphism 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: 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 Rossman (2002). The so-called classical groups generalise 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 cardinality 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 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 group, 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: