If F is a finite field of order q, then we sometimes write GL(n, q) instead of GL(n, F). If the field is R (the real numbers) or C (the complex numbers), the field is sometimes omitted when it is clear from the context, and we write GL(n).
GL(n, F) and subgroups of GL(n, F) are important in the development of group representations, and also arise in the study of spatial symmetries and symmetries of vector spaces in general, as well as the study of polynomials.
If V is a vector space over the field F, then we write GL(V) or Aut(V) for the group of all automorphisms of V, i.e. the set of all bijective linear transformations V → V, together with functional composition as group operation. If the dimension of V is n, then GL(V) and GL(n, F) are isomorphic. The isomorphism is not canonical; it depends on a choice of basis in V. Once a basis has been chosen, every automorphism of V can be represented as an invertible n by n matrix, which establishes the isomorphism.
If n ≥ 2, then the group GL(n, F) is not abelian.
A subgroup of GL(n, F) is called a linear group. Some special subgroups can be identified.
There is the subgroup of all diagonal matrices (all entries except the main diagonal are zero). In fields like R and C, these correspond to rescaling the space; the so called dilations and contractions.
The special linear group, SL(n, F), is the group of all matrices with determinant 1 (that this forms a group follows from the rule of multiplication of determinants). SL(n,F) is in fact a normal subgroup of GL(n,F); and if we write F^{×} for the multiplicative group of F (excluding 0), then
with the isomorphism being induced by the determinant via the first isomorphism theorem.
We can also consider the subgroup of GL(n,F) consisting of all orthogonal matrices, called the orthogonal group O(n, F). In the case F = R, these matrices correspond to automorphisms of R^{n} which respect the Euclidean norm and dot product.
If the field F is R or C, then GL(n) is a Lie group over F of dimension n^{2}. The reason is as follows: GL(n) consists of those matrices whose determinant is nonzero, the determinant is a continuous (even polynomial) map, and hence GL(n) is a nonempty open subset of the manifold of all nbyn matrices, which has dimension n^{2}.
The Lie algebra corresponding to GL(n) consists of all nbyn matrices over F, using the commutator as Lie bracket.
While GL(n,C) is simply connected, GL(n,R) has two connected components: the matrices with positive determinant and the ones with negative determinant. The real nbyn matrices with positive determinant form a subgroup of GL(n,R) denoted by GL^{+}(n,R). This is also a Lie group of real dimension n^{2} and it has the same Lie algebra as GL(n,R). GL^{+}(n,R) is simply connected.
If F is a finite field with q elements, then GL(n, F) is a finite group with
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