Corollary

Let V be a right vector space over the field D, of finite dimension $n\geqslant 2$.

a) The subgroup $\mathbf{S}\mathbf{L}(V)$ of $\mathbf{G}\mathbf{L}(V)$ is generated by the transvections.

b) The subgroup $\mathbf{S}\mathbf{L}(V)$ is the derived group of $\mathbf{G}\mathbf{L}(V)$ unless we have $n= 2$ and D has two elements.

c) The group $\mathbf{S}\mathbf{L}(V)$ is equal to its derived group unless we have $n= 2$ and D has two or three elements.