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

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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

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 \le i,j \le n, i \ne j. The matrix E_{ij}(\lambda) has 1s 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_pq 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

Generalizations to beyond fields

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).