# Elementary matrices of the first kind generate the special linear group over a field

This article states and proves a fact about a particular group, or kind of group, (i.e., special linear group) having a particular generating set (or kind of generating set), i.e., where the generators are elements of the type/form: elementary matrix of the first kind

View other facts about generating sets for particular groups

## Contents

## Statement

Let be a field. Denote by the special linear group over : the group of invertible matrices over with determinant equal to . Then, is generated by elementary matrices (also called *elementary matrices of the first kind* or *shear matrices*), i.e., matrices of the form , where . The matrix has s on the diagonal, in the position, and zeros elsewhere.

The proof is constructive, and shows moreover that the diameter of the Cayley graph using this generating set is at most . In other words, every element of the group can be written as a product of elementary matrices of the first kind using a product length of at most .

### Getting a smaller generating set

Further, since , it suffices to restrict to a subset of that generates additively. Note, however, that if we choose this smaller generating set, the diameter of the Cayley graph could be higher.

## Particular cases

### Finite fields

We consider the case of a field of size with characteristic , and we are interested in studying .

(field size) | (field characteristic) | Number of elementary matrices of the first kind | Diameter of Cayley graph using all elementary matrices of the first kind | Size of generating set using elementary matrices for an additive generating set of the field | Diameter of Cayley graph using such a generating set | ||
---|---|---|---|---|---|---|---|

1 | any | any | any | 0 | 0 | 0 | 0 |

2 | 3 | at most | |||||

2 | 2 | 2 | 1 | 2 | 3 | 2 | 3 |

2 | 3 | 3 | 1 | 4 | 3 | 2 | 3 |

2 | 4 | 2 | 2 | 6 | 3 | 4 | 6 |

2 | 5 | 5 | 1 | 8 | 3 | 2 | 6 |

## Related facts

- Transpositions generate the finitary symmetric group
- 3-cycles generate the finitary alternating group

### Generalizations to beyond fields

- Elementary matrices of the first kind generate the special linear group over a local ring
- Elementary matrices of the first kind generate the special linear group over a Euclidean ring

## Proof

The proof relies on a slight variation of the usual Gauss-Jordan elimination procedure. The variation is necessary because we do not have access to the other types of elementary matrices (permutations and scalar multiplications).