General linear group

In abstract algebra, the general linear group of degree n over a field F (written as GL(n,F)) is the group of n-by-n invertible matrices with entries from F, with the group operation that of ordinary matrix multiplication. (This is indeed a group because the product of two invertible matrices is again invertible.)

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.

Table of contents

1 General linear group of a vector space 2 Subgroups of GL(n, F) 3 Over R and C 4 Over finite fields 5 Projective linear group

General linear group of a vector space

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.

Subgroups of GL(n, F)

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

GL(n,F)/SL(n,F) is isomorphic to F^×

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 Rⁿ which respect the Euclidean norm and dot product.

Over R and C

If the field F is R or C, then GL(n) is a Lie group over F of dimension n². The reason is as follows: GL(n) consists of those matrices whose determinant is non-zero, the determinant is a continuous (even polynomial) map, and hence GL(n) is a non-empty open subset of the manifold of all n-by-n matrices, which has dimension n².

The Lie algebra corresponding to GL(n) consists of all n-by-n 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 n-by-n 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 n² and it has the same Lie algebra as GL(n,R). GL⁺(n,R) is simply connected.

Over finite fields

If F is a finite field with q elements, then GL(n, F) is a finite group with

(qⁿ - 1) · (qⁿ - q) · (qⁿ - q²) · ... · (qⁿ - q^n-1)

elements. This can be shown by counting the possible columns of the matrix: the first column can be anything but the zero column; the second column can be anything but the multiples of the first column, etc.

More generally, one can count points of Grassmannian over F: in other words the number of subspaces of a given dimension k. This requires only finding the order of the stabilizer subgroup of one (described on that page in block matrix form), and divide into the formula just given, by the orbit-stabilizer theorem.

The connection between these formulae, and the Betti numbers of complex Grassmannians, was one of the clues leading to the Weil conjectures.

Projective linear group

The projective linear group of a vector space V over a field K is the quotient group GL(V)/K^x, where K^x consists of the normal subgroup of invertible scalar matrices kI for k in K\\{0}. The notations PGL(V) and so on are analogous to those for the general linear group.

The name comes from projective geometry, where the projective group acting on homogeneous coordinates (x₀:x₁: ... :x_n) is the underlying group of the geometry (NB this is therefore PGL(n+1,K) for projective space of dimension n). The projective linear groups therefore generalise the case PGL₂ of Möbius transformations, sometimes called the Möbius group.