A description of the subgroups of the complete linear group that contain the group of diagonal matrices

Z. I. Borevich

Zapiski Nauchnykh Seminarov POMI, Rings and modules, Tome 64 (1976), pp. 12-29

Résumé

Let $K$ be a field containing at least seven elements. In the group $G=GL(n,K)$ we describe the subgroups containing the group $D$ of all diagonal matrices. This description is given in terms of the concept of a $D$-net subgroup, defined as a subgroup of $G$ composed of matrices $(a_{ij})$ with zero elements $a_{ij}$ in some prescribed cells outside the main diagonal (the set of cells is subordinated to some condition of agreement). The main theorem is: Every subgroup of $G$ containing $D$ is contained between a uniquely determined $D$-net subgroup and its normalizer in $G$. The structure of all subgroups of $G$ containing $D$ is finite and does not depend on the field $K$ (when $\operatorname{card}k\geqslant7$).