# Special linear group

This article defines a natural number-parametrized system of algebraic matrix groups. In other words, for every field and every natural number, we get a matrix group defined by a system of algebraic equations. The definition may also generalize to arbitrary commutative unital rings, though the default usage of the term is over fields.

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## Contents

## Definition

### In terms of natural numbers

Let be a natural number and a field. The **special linear group** of degree over is defined in the following equivalent ways:

- It is the group of all invertible matrices of order , having determinant 1.
- It is the kernel of the determinant homomorphism from the general linear group of degree over to the multiplicative group of .

## More general versions

Replacement of field | Most direct analogue of special linear group | Other less direct analogues |
---|---|---|

Division ring | special linear group over a division ring | |

Commutative unital ring | special linear group over a commutative unital ring | Steinberg group over a commutative unital ring, group generated by elementary matrices over a commutative unital ring |

## As a map

### As a functor from fields to groups

If we fix , we can think of as a functor from the category of fields to the category of groups.

### As an IAPS

`Further information: SL IAPS`

For a fixed field , the general linear groups form an IAPS of groups parametrized by . In other words, we naturally have concatenation maps:

This map takes a matrix of order and a matrix of order and putputs the block diagonal matrix with blocks and .

### As a functor from fields to IAPSes

If we fix neither nor , we can view as a functor from fields to the category of IAPSes of groups.

## Relation with other linear algebraic groups

### Subgroups

- Special orthogonal group
- Symplectic group
- Special orthogonal similitude group
- Unipotent upper-triangular matrix group

### Supergroups

- General linear group
- Special affine group (also called the affine special linear group)
- Outer special linear group
- Special semilinear group

### Group and subgroup operations

- Intersection with the orthogonal group gives the special orthogonal group.
- Intersection with the orthogonal similitude group gives the special orthogonal similitude group.
- Normalizer in the general linear group is the whole general linear group.

## Generating sets and presentations

### Generating sets for the special linear group

- Elementary matrices generate the special linear group: This is true for all special linear groups over all fields.
- Dickson's theorem: This shows that is practically always generated by any upper-triangular unipotent mtarix and any lower-triangular unipotent matrix.

### Presentations for the special linear group

- Steinberg presentation for the special linear group: This is a presentation that works for special linear groups over fields. Over rings that are not fields, the presentation gives a group that has a homomorphism to the special linear group that need not in general be either injective or surjective.

## Particular cases

### Finite fields

Here are some general facts:

- Special linear group is quasisimple for , except the case that and has two or three elements. Thus, all the corresponding general linear groups have a simple non-abelian subquotient. In the case that the field has characteristic two, the general linear group coincides with the special linear group, and it is centerless, so it turns out to be a simple non-abelian group itself.

Size of field | Order of matrices | Common name for the special linear group | Order of group | Comment |
---|---|---|---|---|

1 | Trivial group | Trivial | ||

2 | 2 | Symmetric group:S3 | supersolvable but not nilpotent | |

3 | 2 | Special linear group:SL(2,3) | solvable but not supersolvable | |

4 | 2 | Alternating group:A5 | simple non-abelian group | |

5 | 2 | Special linear group:SL(2,5) | quasisimple, with inner automorphism group isomorphic to alternating group:A5. | |

2 | 3 | General linear group:GL(3,2) | simple non-abelian group |

## GAP implementation

The GAP command for constructing the special linear group is GAP:SpecialLinearGroup. It can be invoked using either `SpecialLinearGroup` or `SL`. It takes two arguments, the first of which is the degree (i.e., the order of matrices) and the second is either a ring or a prime power (for which the corresponding field is considered the ring). For instance, to define , we write:

G := SL(2,3);