General linear groups
Introduction
The general linear group, denoted as GL(n, F), is a fundamental concept in the field of linear algebra and group theory. It consists of all invertible n × n matrices over a given field F, with matrix multiplication serving as the group operation. The general linear group is a cornerstone in the study of vector spaces, as it represents the group of all linear transformations that can be applied to an n-dimensional vector space over F. The structure and properties of general linear groups are pivotal in various branches of mathematics, including algebraic geometry, differential geometry, and representation theory.
Definition and Basic Properties
The general linear group GL(n, F) is defined as the set of all n × n invertible matrices with entries from a field F. A matrix is invertible if there exists another matrix such that their product is the identity matrix. The identity matrix, denoted I_n, is the matrix with ones on the diagonal and zeros elsewhere. The group operation in GL(n, F) is matrix multiplication, which is associative, and the identity element is the identity matrix I_n. The inverse of an element A in GL(n, F) is the matrix A^(-1) such that A * A^(-1) = A^(-1) * A = I_n.
The general linear group is a non-abelian group for n ≥ 2, meaning that the order of multiplication affects the result. This non-commutativity is a key feature that distinguishes GL(n, F) from other groups, such as the symmetric group.
Subgroups and Related Groups
Several important subgroups and related groups arise from the general linear group:
Special Linear Group
The special linear group, denoted SL(n, F), is the subgroup of GL(n, F) consisting of matrices with determinant equal to one. This group is significant in the study of volume-preserving transformations and has applications in physics, particularly in the context of conservation laws.
Orthogonal Group
The orthogonal group, O(n, F), is the subgroup of GL(n, F) consisting of matrices that preserve a quadratic form, typically the Euclidean norm. In the case of real numbers, this group is associated with rotations and reflections in n-dimensional space.
Unitary Group
The unitary group, U(n), is the subgroup of GL(n, C) (where C denotes the complex numbers) consisting of matrices that preserve the Hermitian form. This group is crucial in quantum mechanics and other areas of theoretical physics.
Representations and Applications
General linear groups play a vital role in the representation theory of groups. A representation of a group is a homomorphism from the group to the general linear group of some vector space. This concept allows mathematicians to study abstract groups by representing them as groups of matrices, making them more tangible and easier to analyze.
In algebraic geometry, general linear groups are used to study projective spaces and algebraic varieties. They also appear in the study of Lie groups and Lie algebras, where they serve as examples of matrix Lie groups.
Topological and Geometric Aspects
The general linear group GL(n, F) can be endowed with a topology, making it a topological group. When F is the field of real or complex numbers, GL(n, F) becomes a Lie group, which is a smooth manifold with a group structure. The study of the topology and geometry of GL(n, F) provides insights into the structure of manifolds and fiber bundles.
Algebraic Structure
The algebraic structure of general linear groups is rich and complex. The group GL(n, F) is an example of a linear algebraic group, which is an algebraic group that can be represented as a group of matrices. The study of algebraic groups involves understanding their irreducible representations, characters, and cohomology.
Finite Fields and Finite Groups
When F is a finite field, the general linear group GL(n, F) becomes a finite group. The order of GL(n, F) is given by the formula:
\[ |GL(n, F)| = (q^n - 1)(q^n - q)(q^n - q^2) \cdots (q^n - q^{n-1}) \]
where q is the order of the finite field F. These groups are of particular interest in the study of finite group theory and combinatorics.