The subgroups of the special linear group over a skew field that contain the group of diagonal matrices

Bui Xuan Hai

Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 3, Tome 211 (1994), pp. 91-103

Résumé

For any (noncommutative) skew field $T$, the lattice of subgroups of the special linear group $\Gamma=\mathrm{SL}(n,T)$ that contain the subgroup $\Delta=\mathrm{SD}(n,T)$ of diagonal matrices (with Dieudonné determinants equal to 1) is studied. It is established that for any subgroup $H$, $\Delta\le H\le\Gamma$, there exists a uniquely determined unital net $\sigma$ such that $\Gamma(\sigma)\le H\le\mathcal N(\sigma)$, where $\Gamma(\sigma)$ is the net subgroup associated with the net $\sigma$ and $\mathcal N(\sigma)$ is its normalizer in $\Gamma$. Bibliography: 11 titles.