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
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Statement

Let $K$ be a field. Denote by $SL(n,K)$ the special linear group over $K$: the group of invertible $n\times n$ matrices over $K$ with determinant equal to $1$. Then, $SL(n,K)$ is generated by elementary matrices (also called elementary matrices of the first kind or shear matrices), i.e., matrices of the form $E_{ij}(\lambda )$, where $\lambda \in K,1\leq i,j\leq n,i\neq j$. The matrix $E_{ij}(\lambda )$ has $1$s on the diagonal, $\lambda$ in the $(ij)^{th}$ 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 $n^{2}$. 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 $n^{2}$.

Getting a smaller generating set

Further, since $E_{ij}(\lambda )E_{ij}(\mu )=E_{ij}(\lambda +\mu )$, it suffices to restrict $\lambda$ to a subset of $K$ that generates $K$ 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 $q=p^{r}$ with characteristic $p$, and we are interested in studying $SL(n,q)$.

$n$	$q$ (field size)	$p$ (field characteristic)	$r=\log _{p}q$	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	$q$	$p$	$r$	$2(q-1)$	3	$2r$	at most $3r\lceil (p-1)/2\rceil$
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).