# General linear group

The **general linear group** of degree \( n \) over a ring \( R \) with identity, written \( GL_n(R) \) or \( GL(n,R) \), is the group of all invertible \( n \times n \) matrices with entries in \( R \), under matrix multiplication.<sup>[1](https://mathworld.wolfram.com/GeneralLinearGroup.html)</sup> When \( R \) is a field \( F \), it can equally be described as the set of matrices with nonzero determinant, \( GL_n(F) = \{A \in M_n(F) : \det A \neq 0\} \), or as the group of invertible linear transformations of the vector space \( F^n \).<sup>[1](https://mathworld.wolfram.com/GeneralLinearGroup.html)</sup> It is the automorphism group \( \mathrm{Aut}_K(V) \) of a free module of rank \( n \), and it serves as the ambient group in which the classical groups, such as the special, orthogonal, and unitary groups, live as subgroups.<sup>[2](https://encyclopediaofmath.org/wiki/General_linear_group)</sup><sup> • </sup><sup>[3](https://www.math.ucla.edu/~rse/algebra_book.pdf)</sup>

| Key fact | Detail |
|---|---|
| Definition | \( GL_n(R) \) is the group of units of the matrix ring \( M_n(R) \); over a field, the matrices with nonzero determinant<sup>[1](https://mathworld.wolfram.com/GeneralLinearGroup.html)</sup><sup> • </sup><sup>[3](https://www.math.ucla.edu/~rse/algebra_book.pdf)</sup>  |
| Determinant map | \( \det : GL_n(F) \to F^{\times} \) is a surjective group homomorphism with kernel \( SL_n(F) \), giving \( 1 \to SL_n(F) \to GL_n(F) \to F^{\times} \to 1 \)<sup>[1](https://mathworld.wolfram.com/GeneralLinearGroup.html)</sup> |
| Center | Over a field, the nonzero scalar matrices \( \lambda I \); the quotient is the projective general linear group \( PGL_n \)<sup>[1](https://mathworld.wolfram.com/GeneralLinearGroup.html)</sup> |
| Order over \( \mathbb{F}_q \) | \( |GL(n,q)| = (q^n - 1)(q^n - q) \cdots (q^n - q^{n-1}) \), and \( |PGL(n,q)| = |GL(n,q)|/(q-1) \)<sup>[4](https://webspace.maths.qmul.ac.uk/p.j.cameron/class_gps/ch2.pdf)</sup> |
| Simplicity | For a field \( K \), \( PSL(n,K) = SL(n,K)/(SL(n,K) \cap Z_n) \) is simple apart from \( n = 2 \), \( |K| = 2, 3 \)<sup>[2](https://encyclopediaofmath.org/wiki/General_linear_group)</sup> |
| Lie group structure | \( GL(n,\mathbb{R}) \) is a real Lie group of dimension \( n^2 \)<sup>[5](https://resources.saylor.org/wwwresources/archived/site/wp-content/uploads/2011/04/General-linear-group.pdf)</sup> |
| Integer case | \( GL_n(\mathbb{Z}) \) consists of integer matrices with \( \det(A) = \pm 1 \)<sup>[6](https://www.math.ucla.edu/~sharifi/notes/algebra-ch04.html)</sup> |

## Definition over rings and fields

For any ring \( R \), the units of the full matrix ring \( M_n(R) \) form a group under multiplication, and this group is \( GL_n(R) \).<sup>[3](https://www.math.ucla.edu/~rse/algebra_book.pdf)</sup> If \( R \) is commutative, the determinant still makes sense, and a matrix \( A \) is a unit if and only if \( \det(A) \in R^{\times} \), the group of units of the ring.<sup>[3](https://www.math.ucla.edu/~rse/algebra_book.pdf)</sup> Over a field this condition reduces to a nonzero determinant, since \( R^{\times} = R \setminus \{0\} \); over a commutative ring that is not a field, an element can fail to be a unit without being zero, so the two conditions differ.<sup>[7](https://math.stackexchange.com/questions/296229/definition-of-textgln-r)</sup>

The same group can be defined without coordinates: \( GL(n,K) \) is the automorphism group of the free right \( K \)-module \( V \) with \( n \) generators, and choosing a basis identifies it with invertible matrices.<sup>[2](https://encyclopediaofmath.org/wiki/General_linear_group)</sup><sup> • </sup><sup>[4](https://webspace.maths.qmul.ac.uk/p.j.cameron/class_gps/ch2.pdf)</sup>  For \( n > 1 \) the group is not Abelian.<sup>[8](https://webspace.maths.qmul.ac.uk/p.j.cameron/notes/algstr.pdf)</sup>

## The determinant and the special linear group

The determinant is a surjective group homomorphism from \( GL_n(F) \) to the multiplicative group \( F^{\times} \).<sup>[1](https://mathworld.wolfram.com/GeneralLinearGroup.html)</sup> Its kernel is the special linear group \( SL_n(F) \), the matrices of determinant 1, so there is an [exact sequence](https://www.edgepedia.org/exact-sequence)

\[ 1 \to SL_n(F) \to GL_n(F) \xrightarrow{\det} F^{\times} \to 1. \]

Over a commutative ring \( R \), \( SL_n(R) = \{A \in GL_n(R) \mid \det A = 1\} \) is defined the same way.<sup>[1](https://mathworld.wolfram.com/GeneralLinearGroup.html)</sup><sup> • </sup><sup>[3](https://www.math.ucla.edu/~rse/algebra_book.pdf)</sup>  In the integer case, an \( n \times n \) matrix with integer entries has an inverse with integer entries if and only if its determinant is \( \pm 1 \), which characterizes \( GL_n(\mathbb{Z}) \); the determinant map then has image \( \{\pm 1\} \), so \( [GL(n,\mathbb{Z}) : SL(n,\mathbb{Z})] = 2 \).<sup>[6](https://www.math.ucla.edu/~sharifi/notes/algebra-ch04.html)</sup><sup> • </sup><sup>[2](https://encyclopediaofmath.org/wiki/General_linear_group)</sup>

## Center and projective groups

The center of \( GL_n(F) \) consists of the nonzero scalar matrices \( \lambda I \), and factoring by this center gives the projective general linear group \( PGL_n(F) \).<sup>[1](https://mathworld.wolfram.com/GeneralLinearGroup.html)</sup> Geometrically, \( GL(n,F) \) acts on the projective space \( PG(n-1,F) \) because an invertible linear transformation maps a subspace to another subspace of the same dimension, and the kernel of this action on points is exactly the subgroup of central scalar matrices \( \{cI : c \in Z(F),\ c \neq 0\} \), where \( Z(F) \) is the center of the division ring \( F \).<sup>[4](https://webspace.maths.qmul.ac.uk/p.j.cameron/class_gps/ch2.pdf)</sup> The image of \( SL_n(F) \) in this quotient is the projective special linear group \( PSL(n,F) \).<sup>[4](https://webspace.maths.qmul.ac.uk/p.j.cameron/class_gps/ch2.pdf)</sup>

## Normal structure and simplicity

The normal structure of \( GL(n,K) \) is a central topic in its study and is connected with algebraic \( K \)-theory.<sup>[2](https://encyclopediaofmath.org/wiki/General_linear_group)</sup> For a field \( K \), the commutator subgroup of \( GL(n,K) \) is identical to \( SL(n,K) \) except when \( n = 2 \) and \( |K| = 2 \); moreover, every normal subgroup of \( GL(n,K) \) either lies inside the center \( Z_n \) or includes \( SL(n,K) \).<sup>[2](https://encyclopediaofmath.org/wiki/General_linear_group)</sup> Consequently \( PSL(n,K) \) is a simple group apart from the cases \( n = 2 \), \( |K| = 2, 3 \).<sup>[2](https://encyclopediaofmath.org/wiki/General_linear_group)</sup> Over a skew-field with \( n > 1 \), the analogous statements hold with the commutator subgroup \( SL^+(n,K) \) generated by transvections, and \( GL(n,K)/SL^+(n,K) \simeq K^*/[K^*,K^*] \).<sup>[2](https://encyclopediaofmath.org/wiki/General_linear_group)</sup>

Over a general ring the picture changes: \( GL(n,K) \) may contain numerous normal subgroups. Suppose \( K \) is a finitely generated integral domain; in that case, the group \( GL(n,K) \) is residually finite—that is, given any element \( g \), one can find a normal subgroup \( N_g \) of finite index that excludes \( g \).<sup>[2](https://encyclopediaofmath.org/wiki/General_linear_group)</sup>

## Finite fields and order formulas

Over the finite field \( GF(q) \), the group is written \( GL(n,q) \), and its order is counted column by column: an [invertible matrix](https://www.edgepedia.org/invertible-matrix) has linearly independent rows, giving

\[ |GL(n,q)| = (q^n - 1)(q^n - q) \cdots (q^n - q^{n-1}), \]

and \( |PGL(n,q)| = |GL(n,q)|/(q-1) \).<sup>[4](https://webspace.maths.qmul.ac.uk/p.j.cameron/class_gps/ch2.pdf)</sup> These finite groups supply many of the finite simple groups through their projective special subgroups.<sup>[4](https://webspace.maths.qmul.ac.uk/p.j.cameron/class_gps/ch2.pdf)</sup><sup> • </sup><sup>[9](https://webspace.maths.qmul.ac.uk/r.a.wilson/FSG2008/simplegroups.html)</sup>

## Generation

Generation depends on the coefficient ring. For an infinite field \( F \), \( GL_n(F) \) is not finitely generated: the determinants of any generating set would finitely generate \( F^{\times} \), but \( F^{\times} \) contains a copy of \( \mathbb{Q}^* \), of \( \mathbb{F}_p(T)^* \), or is torsion but infinite, so it cannot be finitely generated.<sup>[10](https://math.stackexchange.com/questions/2654172/general-linear-group-gl-nr-not-finitely-generated)</sup> Over an infinite ring the situation can differ; \( GL_n(\mathbb{Z}) \) is finitely generated.<sup>[10](https://math.stackexchange.com/questions/2654172/general-linear-group-gl-nr-not-finitely-generated)</sup>

## Topology and Lie theory

Over \( \mathbb{R} \) or \( \mathbb{C} \), invertibility is equivalent to a nonvanishing determinant, so \( GL_n \) is an open subset of the space of all matrices.<sup>[11](https://www.math.ucla.edu/~petersen/manifolds.pdf)</sup> With this structure \( GL(n,\mathbb{R}) \) is a real [Lie group](https://www.edgepedia.org/lie-group) of dimension \( n^2 \), and it and its closed subgroups are the standard examples of linear Lie groups, whose Lie algebras sit inside the general matrix [Lie algebra](https://www.edgepedia.org/lie-algebra).<sup>[5](https://resources.saylor.org/wwwresources/archived/site/wp-content/uploads/2011/04/General-linear-group.pdf)</sup><sup> • </sup><sup>[12](https://encyclopediaofmath.org/wiki/Lie_group)</sup>

## Subgroups

Over a commutative ring \( R \), the standard subgroups of \( GL_n(R) \) include the special linear group \( SL_n(R) \), the orthogonal group \( O_n(R) = \{A \mid A^t A = I\} \) with its intersection \( SO_n(R) = SL_n(R) \cap O_n(R) \), the diagonal group \( D_n(R) \), the upper triangular group \( T_n(R) \) with its strictly upper triangular subgroup, and, over \( \mathbb{C} \), the unitary group \( U_n(C) = \{A \mid A^* A = I\} \) and the special unitary group \( SU_n(C) = SL_n(C) \cap U_n(C) \).<sup>[3](https://www.math.ucla.edu/~rse/algebra_book.pdf)</sup> The symplectic group \( Sp_{2n} \) is defined inside \( GL_{2n} \) by the condition \( A^t J A = J \) for a fixed skew matrix \( J \), and the [Lorentz group](https://www.edgepedia.org/lorentz-group) \( O_{3,1}(R) \) inside \( GL_4(\mathbb{R}) \) by preservation of the form \( \mathrm{diag}(1,1,1,-1) \).<sup>[3](https://www.math.ucla.edu/~rse/algebra_book.pdf)</sup>

## Role in representation theory

[Representation theory](https://www.edgepedia.org/representation-theory) is the branch of mathematics that represents the elements of an abstract group as elements of a general linear group of a chosen vector space by means of a homomorphism, mapping group elements to linear operators so the group can be studied with linear algebra.<sup>[13](https://digitalcommons.colby.edu/honorstheses/1370/)</sup> In this sense \( GL(V) \) is the target of every linear representation, and its subgroups carry the representations used in physics, where Lie group representations underlie applications in quantum physics.<sup>[13](https://digitalcommons.colby.edu/honorstheses/1370/)</sup> The group action of a general linear group on tensors has also been studied as a candidate framework for post-quantum cryptography, through the hardness of the associated tensor isomorphism problem.<sup>[14](https://arxiv.org/abs/1906.04330)</sup>

## References

1. [General Linear Group, Wolfram MathWorld](https://mathworld.wolfram.com/GeneralLinearGroup.html)
2. [General linear group, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/General_linear_group)
3. [R. E. Sharpley (ed. notes), Algebra book, UCLA, ch. III: Groups](https://www.math.ucla.edu/~rse/algebra_book.pdf)
4. [P. J. Cameron, Classical Groups, ch. 2: Linear and projective groups, Queen Mary University of London](https://webspace.maths.qmul.ac.uk/p.j.cameron/class_gps/ch2.pdf)
5. [General linear group (Saylor archived text)](https://resources.saylor.org/wwwresources/archived/site/wp-content/uploads/2011/04/General-linear-group.pdf)
6. [Romyar Sharifi, Abstract Algebra, ch. 4: Advanced group theory, UCLA](https://www.math.ucla.edu/~sharifi/notes/algebra-ch04.html)
7. [Definition of GL(n,R), Mathematics Stack Exchange](https://math.stackexchange.com/questions/296229/definition-of-textgln-r)
8. [P. J. Cameron, Notes on Algebraic Structures, Queen Mary University of London](https://webspace.maths.qmul.ac.uk/p.j.cameron/notes/algstr.pdf)
9. [R. A. Wilson, Finite Simple Groups, Queen Mary University of London](https://webspace.maths.qmul.ac.uk/r.a.wilson/FSG2008/simplegroups.html)
10. [General Linear Group GL_n(R) not Finitely Generated, Mathematics Stack Exchange](https://math.stackexchange.com/questions/2654172/general-linear-group-gl-nr-not-finitely-generated)
11. [K. Petersen, Manifold Theory, UCLA](https://www.math.ucla.edu/~petersen/manifolds.pdf)
12. [Lie group, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Lie_group)
13. [J. Bystrický, Representation Theory and its Applications in Physics, Colby College Honors Theses (2022)](https://digitalcommons.colby.edu/honorstheses/1370/)
14. [General Linear Group Action on Tensors: A Candidate for Post-Quantum Cryptography, arXiv:1906.04330](https://arxiv.org/abs/1906.04330)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Linear and multilinear algebra › Matrix theory › Matrices over rings, fields, and other structures*

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