# Special linear group

The **special linear group** is, for a commutative ring \( R \) with identity, the group of \( n \times n \) matrices with entries in \( R \) and determinant 1, written \( \mathrm{SL}_n(R) \) or \( \mathrm{SL}(n, R) \). Over a field \( F \) it is the subgroup of the general linear group \( \mathrm{GL}_n(F) \) consisting of the invertible matrices of determinant exactly 1: \( \mathrm{SL}_n(F) = \{ A \in \mathrm{GL}_n(F) : \det A = 1 \} \)<sup>[1](https://mathworld.wolfram.com/SpecialLinearGroup.html)</sup>. It is one of the classical matrix groups and, in its projective form \( \mathrm{PSL}_n \), supplies an infinite family of finite simple groups.<sup>[2](https://kconrad.math.uconn.edu/blurbs/grouptheory/PSLnsimple.pdf)</sup>

| Key fact | Detail |
|---|---|
| Definition | \( \mathrm{SL}_n(R) \) = all \( n \times n \) matrices over \( R \) with determinant 1; over a field, \( \mathrm{SL}_n(F) = \{ A \in \mathrm{GL}_n(F) : \det A = 1 \} \)<sup>[1](https://mathworld.wolfram.com/SpecialLinearGroup.html)</sup> |
| Structural position | Kernel of the determinant homomorphism \( \det : \mathrm{GL}_n(F) \to F^{\times} \), hence a normal subgroup of \( \mathrm{GL}_n(F) \)<sup>[1](https://mathworld.wolfram.com/SpecialLinearGroup.html)</sup> |
| Order over \( \mathbb{F}_q \) | \( |\mathrm{SL}_n(q)| = |\mathrm{GL}_n(q)|/(q-1) = q^{n(n-1)/2} \prod_{j=2}^{n} (q^j - 1) \)<sup>[1](https://mathworld.wolfram.com/SpecialLinearGroup.html)</sup> |
| Center | Scalar matrices \( \lambda I \) with \( \lambda^n = 1 \); over \( \mathbb{F}_q \) the center has order \( d = \gcd(n, q-1) \)<sup>[1](https://mathworld.wolfram.com/SpecialLinearGroup.html)</sup><sup> • </sup><sup>[3](https://mathworld.wolfram.com/ProjectiveSpecialLinearGroup.html)</sup> |
| Simplicity | \( \mathrm{PSL}_n(F) = \mathrm{SL}_n(F)/Z(\mathrm{SL}_n(F)) \) is simple for all \( n \ge 2 \) and all fields \( F \), except \( n = 2 \) with \( F = \mathbb{F}_2 \) and \( \mathbb{F}_3 \)<sup>[2](https://kconrad.math.uconn.edu/blurbs/grouptheory/PSLnsimple.pdf)</sup> |
| Other names | Unimodular group; over a skew-field \( K \) it equals the subgroup of \( \mathrm{GL}_n(K) \) generated by all transvections<sup>[4](https://encyclopediaofmath.org/wiki/Classical_group)</sup> |
| Geometry | \( \mathrm{SL}_2(\mathbb{R}) \) acts on the upper half-plane by \( z \mapsto (az+b)/(cz+d) \), the action underlying the modular group \( \mathrm{SL}_2(\mathbb{Z}) \)<sup>[5](https://metaphor.ethz.ch/x/2022/fs/401-4118-22L/notes/MF_lectures_1-8.pdf)</sup> |

## Definition and basic structure

The determinant is multiplicative, so the set of matrices with determinant 1 is closed under multiplication and inversion and forms a subgroup of the general linear group.<sup>[6](https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Abstract_Algebra%3A_Theory_and_Applications_%28Judson%29/12%3A_Matrix_Groups_and_Symmetry/12.01%3A_Matrix_Groups)</sup> More precisely, the determinant map \( \det : \mathrm{GL}_n(F) \to F^{\times} \) is a group homomorphism to the multiplicative group of the field, and \( \mathrm{SL}_n(F) \) is its kernel, the preimage of the identity element 1. Kernels of homomorphisms are normal subgroups, so \( \mathrm{SL}_n(F) \triangleleft \mathrm{GL}_n(F) \), and the first isomorphism theorem identifies the quotient \( \mathrm{GL}_n(F)/\mathrm{SL}_n(F) \) with \( F^{\times} \).<sup>[1](https://mathworld.wolfram.com/SpecialLinearGroup.html)</sup>

The same construction works over any commutative ring with identity, not only fields, and the Encyclopedia of Mathematics records an equivalent characterization over a skew-field \( K \): \( \mathrm{SL}_n(K) \) is the subgroup of \( \mathrm{GL}_n(K) \) generated by all transvections, and it coincides with the automorphisms of determinant 1. For this reason the group is also called the unimodular group.<sup>[4](https://encyclopediaofmath.org/wiki/Classical_group)</sup> A transvection is a linear map that fixes a hyperplane pointwise and shifts vectors parallel to it; these are the elementary row-and-column operations of linear algebra in group-theoretic clothing.

Geometrically, the determinant measures how a linear map scales oriented volume. A matrix with determinant \( \pm 1 \) preserves size, one with positive determinant preserves orientation, and one with determinant exactly 1 preserves both.<sup>[7](https://cs.kingsu.ca/~remkes/notes/linearalgebra/section-matrix-groups.html)</sup> Thus \( \mathrm{SL}_n(\mathbb{R}) \) is the group of volume-preserving, orientation-preserving linear changes of coordinates of \( n \)-dimensional real space.

## Relation to other classical groups

The special linear group sits inside a family of matrix groups defined by determinant or form conditions. The orthogonal group \( O(n) \) consists of matrices preserving the dot product; its determinant-one part, the [special orthogonal group](https://www.edgepedia.org/special-orthogonal-group) \( SO(n) \), is exactly the intersection \( O(n) \cap \mathrm{SL}_n(\mathbb{R}) \).<sup>[6](https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Abstract_Algebra%3A_Theory_and_Applications_%28Judson%29/12%3A_Matrix_Groups_and_Symmetry/12.01%3A_Matrix_Groups)</sup> Over \( \mathbb{C} \), the unitary group \( U(n) \subset \mathrm{GL}(n, \mathbb{C}) \) has its determinant-one subgroup \( SU(n) \), the special unitary group.<sup>[8](https://sites.ualberta.ca/~vbouchar/MAPH464/section-subgroup.html)</sup> In each case the adjective "special" signals the determinant-one condition.

Over the real numbers the group \( \mathrm{SL}_2(\mathbb{R}) \) is a connected, non-compact simple [Lie group](https://www.edgepedia.org/lie-group) of dimension 3, with applications in geometry, topology, representation theory, and physics.<sup>[9](https://en.wiki.beta.math.wmflabs.org/wiki/SL2%28R%29)</sup>

## Finite fields and group order

When the ground field is the finite field \( \mathbb{F}_q \), where \( q \) is a prime power, the group is finite and its order has a closed formula:

\[ |\mathrm{SL}_n(q)| = \frac{|\mathrm{GL}_n(q)|}{q-1} = q^{n(n-1)/2} \prod_{j=2}^{n} (q^j - 1). \]

The division by \( q - 1 \) reflects the determinant map: among the invertible matrices, exactly one fraction in \( q-1 \) has each possible nonzero determinant value, so restricting to determinant 1 removes a factor of \( q-1 \).<sup>[1](https://mathworld.wolfram.com/SpecialLinearGroup.html)</sup> For example, \( |\mathrm{SL}_2(\mathbb{F}_7)| = (7^2-1)(7^2-7)/(7-1) \cdot \) adjustment gives 336, and the projective quotient has order 168 (see below).

The center of \( \mathrm{SL}_n \) consists of the scalar matrices \( \lambda I \) for which \( \lambda^n = 1 \).<sup>[1](https://mathworld.wolfram.com/SpecialLinearGroup.html)</sup> Over \( \mathbb{F}_q \) this center has order \( d = \gcd(n, q-1) \), since the nonzero elements of \( \mathbb{F}_q \) form a cyclic group of order \( q-1 \) and the solutions of \( \lambda^n = 1 \) inside it number \( \gcd(n, q-1) \).<sup>[3](https://mathworld.wolfram.com/ProjectiveSpecialLinearGroup.html)</sup>

## The projective special linear group and simplicity

Quotienting \( \mathrm{SL}_n(F) \) by its center gives the **projective special linear group**

\[ \mathrm{PSL}_n(F) = \mathrm{SL}_n(F) / Z(\mathrm{SL}_n(F)). \]

This quotient is the central object in the theory of the classical groups because of a simplicity theorem: \( \mathrm{PSL}_n(F) \) is a simple group for all \( n \ge 2 \) and all fields \( F \), with the sole exceptions \( n = 2 \) over \( \mathbb{F}_2 \) and \( \mathbb{F}_3 \), where \( \mathrm{PSL}_2(\mathbb{F}_2) \cong S_3 \) and \( \mathrm{PSL}_2(\mathbb{F}_3) \cong A_4 \), neither of which is simple.<sup>[2](https://kconrad.math.uconn.edu/blurbs/grouptheory/PSLnsimple.pdf)</sup> A thesis from [Lund University](https://www.edgepedia.org/lund-university) states the same result in the form that \( \mathrm{PSL}(m, K) \) is simple for all dimensions \( m \) and finite fields \( K \), with a few exceptions.<sup>[10](https://www.lunduniversity.lu.se/publication/8935153)</sup>

The theorem has a history. In 1831 Galois claimed, without proof, that \( \mathrm{PSL}_2(\mathbb{F}_p) \) is simple for all primes \( p > 3 \). The full result was developed over roughly 30 years, from 1870 to 1901: [Camille Jordan](https://www.edgepedia.org/camille-jordan) handled \( n \ge 2 \) over prime fields except \( (n,p) = (2,2) \) and \( (2,3) \); Eliakim H. Moore, an American mathematician who was a founder of American research mathematics and a president of the American Mathematical Society, covered \( n = 2 \) over all finite fields of size greater than 3; and Leonard Dickson completed the cases \( n > 2 \) over finite fields and \( n \ge 2 \) over infinite fields.<sup>[2](https://kconrad.math.uconn.edu/blurbs/grouptheory/PSLnsimple.pdf)</sup> A modern proof by Keith Conrad, a mathematician at the [University of Connecticut](https://www.edgepedia.org/university-of-connecticut), uses a 1941 criterion of Iwasawa relating simple quotient groups to doubly transitive group actions; the action of \( \mathrm{SL}(n, F) \) on projective space is doubly transitive for \( n \ge 2 \).<sup>[2](https://kconrad.math.uconn.edu/blurbs/grouptheory/PSLnsimple.pdf)</sup><sup> • </sup><sup>[11](https://www.people.brandeis.edu/~igusa/Math131b/PSL.pdf)</sup>

The smallest non-abelian simple group is \( A_5 \); the next one up to isomorphism is \( \mathrm{PSL}_2(\mathbb{F}_7) \), of order 168.<sup>[12](https://www.math.ucla.edu/~rse/algebra_book.pdf)</sup> Since \( |Z(\mathrm{SL}_2(\mathbb{F}_7))| = \gcd(2, 6) = 2 \), this group is the quotient of \( \mathrm{SL}_2(\mathbb{F}_7) \), of order 336, by its center \( \{ \pm I \} \).<sup>[3](https://mathworld.wolfram.com/ProjectiveSpecialLinearGroup.html)</sup>

## Action on the upper half-plane and the modular group

The real group \( \mathrm{SL}_2(\mathbb{R}) \) acts on the upper half-plane \( \mathbb{H} = \{ z \in \mathbb{C} : \Im(z) > 0 \} \) by fractional linear transformations,

\[ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \cdot z = \frac{az + b}{cz + d}. \]

A direct computation shows this map sends \( \mathbb{H} \) to itself, with \( \Im\left( \frac{az+b}{cz+d} \right) = \frac{\Im(z)}{|cz+d|^2} \) when \( \det A = 1 \), and that the action respects composition.<sup>[5](https://metaphor.ethz.ch/x/2022/fs/401-4118-22L/notes/MF_lectures_1-8.pdf)</sup>

Restricting to integer entries gives the **modular group** \( \mathrm{SL}_2(\mathbb{Z}) \), the matrices \( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) with \( ad - bc = 1 \) and \( a, b, c, d \in \mathbb{Z} \).<sup>[13](https://wstein.org/books/modform/modform/modform.html)</sup> The quotient of the upper half-plane by \( \mathrm{SL}_2(\mathbb{Z}) \) is a [Riemann surface](https://www.edgepedia.org/riemann-surface), and \( \mathrm{SL}_2(\mathbb{Z}) \) is generated by two matrices \( S \) and \( T \) corresponding to the transformations \( z \mapsto -1/z \) and \( z \mapsto z + 1 \).<sup>[14](https://sites.math.duke.edu/~hain/teaching/mth790/)</sup> This group and its subgroups organize the theory of modular forms, whose symmetry conditions are stated in terms of the \( \mathrm{SL}_2 \) action on \( \mathbb{H} \).<sup>[5](https://metaphor.ethz.ch/x/2022/fs/401-4118-22L/notes/MF_lectures_1-8.pdf)</sup>

## Algebraic characterizations

Two further facts locate \( \mathrm{SL}_n \) within \( \mathrm{GL}_n \). And over the integers, \( \mathrm{SL}_n(\mathbb{Z}) \) acts on tori: for any irreducible hyperbolic automorphism \( \phi \in \mathrm{SL}_n(\mathbb{Z}) \) of the \( n \)-torus with \( n \ge 4 \), there exist pseudo-Anosov mappings semiconjugate and almost-isomorphic to sufficiently large powers of \( \phi \), a result connecting the group to surface dynamics.<sup>[15](https://arxiv.org/abs/1912.09385)</sup>

## Computation

The group is directly available in computer algebra systems. GAP, a system for computational discrete algebra with emphasis on computational group theory, provides general and special linear groups as built-in objects; SageMath wraps these through its class `LinearMatrixGroup_gap`, which constructs "the general or special linear group in GAP" from a degree, a base ring, and a flag recording whether the group is special.<sup>[16](https://doc.sagemath.org/html/en/reference/spkg/gap.html)</sup><sup> • </sup><sup>[17](https://doc.sagemath.org/html/en/reference/groups/sage/groups/matrix_gps/linear_gap.html)</sup> Sage also exposes GAP through an interpreter interface supporting expression evaluation and Pythonic wrappers of GAP objects.<sup>[18](https://doc.sagemath.org/html/en/reference/interfaces/sage/interfaces/gap.html?highlight=gap4)</sup>

## References

1. [Special Linear Group, Wolfram MathWorld](https://mathworld.wolfram.com/SpecialLinearGroup.html)
2. [Keith Conrad, Simplicity of PSL_n(F), University of Connecticut](https://kconrad.math.uconn.edu/blurbs/grouptheory/PSLnsimple.pdf)
3. [Projective Special Linear Group, Wolfram MathWorld](https://mathworld.wolfram.com/ProjectiveSpecialLinearGroup.html)
4. [Classical group, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Classical_group)
5. [Modular Forms, lecture notes, ETH Zürich (2022)](https://metaphor.ethz.ch/x/2022/fs/401-4118-22L/notes/MF_lectures_1-8.pdf)
6. [Matrix Groups, Abstract Algebra: Theory and Applications, LibreTexts](https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Abstract_Algebra%3A_Theory_and_Applications_%28Judson%29/12%3A_Matrix_Groups_and_Symmetry/12.01%3A_Matrix_Groups)
7. [Matrix Groups, linear algebra notes, The King's University](https://cs.kingsu.ca/~remkes/notes/linearalgebra/section-matrix-groups.html)
8. [Subgroups, MAPH 464 notes, University of Alberta](https://sites.ualberta.ca/~vbouchar/MAPH464/section-subgroup.html)
9. [SL2(R), testwiki](https://en.wiki.beta.math.wmflabs.org/wiki/SL2%28R%29)
10. [On Simplicity of the Projective Special Linear Groups, Lund University](https://www.lunduniversity.lu.se/publication/8935153)
11. [Definitions on PSL, Brandeis University course notes](https://www.people.brandeis.edu/~igusa/Math131b/PSL.pdf)
12. [Algebra notes, UCLA (R. S. Enns et al.)](https://www.math.ucla.edu/~rse/algebra_book.pdf)
13. [Modular Forms, A Computational Approach, W. Stein](https://wstein.org/books/modform/modform/modform.html)
14. [Math 790, Fall 2021, Duke University](https://sites.math.duke.edu/~hain/teaching/mth790/)
15. [Pseudo-Anosov mappings and toral automorphisms, arXiv:1912.09385](https://arxiv.org/abs/1912.09385)
16. [GAP: Groups, Algorithms, Programming, SageMath package documentation](https://doc.sagemath.org/html/en/reference/spkg/gap.html)
17. [Linear Groups with GAP, SageMath reference](https://doc.sagemath.org/html/en/reference/groups/sage/groups/matrix_gps/linear_gap.html)
18. [Interface to GAP, SageMath reference](https://doc.sagemath.org/html/en/reference/interfaces/sage/interfaces/gap.html?highlight=gap4)

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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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