Some presentations for the special linear groups on finite fields. (Q1420362)

The so-called Steinberg presentation of the finite group \(\text{SL}(n,q)\), \(q=p^m\) a power of the odd prime \(p\) is a presentation of \(\text{SL}(n,q)\) as an abstract group with generators \(x_{ij}(\alpha)\), \(1\leq i,j\leq n\), \(i\neq j\), \(\alpha\in\text{GF}(q)\) together with the relations (E1) \(x_{ij}(\alpha)x_{ij}(\beta )=x_{ij}(\alpha+\beta)\), (E2) \([x_{ij}(\alpha),x_{k\ell}(\beta)]=1\) for \(1\leq i,j,k,\ell\leq n\), \(i\neq\ell\), \(j\neq k\), (E3) \([x_{ij}(\alpha),x_{jk}(\beta)]=x_{ik}(\alpha\beta)\), \(1\leq i,j,k\leq n\), \(i\neq j\neq k\neq i\), and \(\alpha,\beta\in\text{GF}(q)\). The author shows that these relations can be replaced by the same relations where \(\alpha\) and \(\beta\) range over the elements of a \(\text{GF}(p)\)-basis of \(\text{GF}(q)\) of the form \(\{1,\gamma,\gamma^2,\dots,\gamma^{m-1}\}\). Thus the number of relations is reduced considerably.