Subnormalizer of net subgroups in the general linear group over a ring

Z. I. Borevich ; L. Yu. Kolotilina

Zapiski Nauchnykh Seminarov POMI, Integral lattices and finite linear groups, Tome 116 (1982), pp. 14-19

Résumé

Let $\Lambda$ be a commutative ring in which the elements of the form $\varepsilon^2-1$, $\varepsilon\in\Lambda^*$ generate the unit ideal and assume that $\sigma$ is any $D$-net of ideals of $\Lambda$ of order $n$. It is shown that the normalizer $N(\sigma)$ of the net subgroup $G(\sigma)$ (RZhMat, 1977, 2A280) coincides with its subnormalizer in $GL(n,\Lambda)$. For noncommutative $\Lambda$ the corresponding result is obtained under the assumptions: 1) in $\Lambda$ the elements of the form $\varepsilon-1$, where $\varepsilon$ runs through all invertible elements of the center of $\Lambda$, generate the unit ideal, and 2) the subgroup $G(\sigma)$ contains the group of block diagonal matrices with blocks of order $\geqslant2$.