On the Diagonal Subgroup of the Special Linear Group Over a Division Ring

Let K be a division ring with center Z(K), and n a positive integer. Let \(\textrm{SL}(n,K)\) $\text{SL}(n,K)$ be the special linear group of degree n over K and \(\textrm{SD}(n,K)\) $\text{SD}(n,K)$ its subgroup consisting of all diagonal matrices whose Dieudonne’s determinant is \(\overline{1}\) $\overline{1}$ . We prove that \(\textrm{SD}(n,K)\) $\text{SD}(n,K)$ is weakly pronormal, but not pronormal in \(\textrm{SL}(n,K)\) $\text{SL}(n,K)$ provided either Z(K) is an infinite field in case \(n\ge 3\) $n\ge 3$ or Z(K) is a finite field containing at least seven elements in case \(n\ge 5\) $n\ge 5$ .