On the diagonal subgroup of the special linear group over a division ring (Q6650706)

Let \(K\) be a division ring with center \(Z(K)\), \(n\) a positive integer, \(K^{\ast}\) the multiplicative group of \(K\) and \(K'=[K^{\ast}, K^{\ast}]\). The description of subgroups of the special linear group \(\Gamma=\mathrm{SL}(n,K)\) containing the diagonal subgroup \(\Delta=\mathrm{SD}(n, K)\) was obtained by the author in [J. Sov. Math. 57, No. 6, 3449--3452 (1991; Zbl 0791.20053)] for \(n \geq 3\) if \(Z(K)\) is an infinite field, and in [\textit{Bui Xuan Hai}, J. Pure Appl. Algebra 121, No. 1, 53--67 (1997; Zbl 0881.20023)], for \(n \geq 5\) if \(Z(K)\) is a finite field containing at least seven elements.\N\NIn the paper under review the author proves that \(\mathrm{SD}(n, K)\) is weakly pronormal, but not pronormal in \(\mathrm{SL}(n, K)\) provided either \(Z(K)\) is an infinite field in case \(n \geq 3\) or \(Z(K)\) is a finite field containing at least seven elements in case \(n \geq 5\).