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
.
![]() |
![]() |
![]() |
![]() |
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).