Location of subgroups in the general linear group over a commutative ring
For an arbitrary commutative ring R, the authors study the lattice of subgroups of the general linear group \(G=GL(n,R)\) that contain a fixed elementary block-diagonal group E(\(\nu)\) of given type \(\nu\), subject to the condition that all the diagonal blocks in E(\(\nu)\) be of degree \(\geq 3\). For a D-net \(\sigma\) of ideals in R of degree n, denote the corresponding net subgroup by G(\(\sigma)\), its normalizer in G by N(\(\sigma)\), and by E(\(\sigma)\) the subgroup generated by the elementary transvections in G(\(\sigma)\). Main result: For every intermediate subgroup H, E(\(\nu)\)\(\leq H\leq G\), there is a uniquely determined D-net \(\sigma\) such that E(\(\sigma)\)\(\leq H\leq N(\sigma)\). In addition, E(\(\sigma)\) is a normal subgroup of N(\(\sigma)\), and N(\(\sigma)\) coincides with the subnormalizer of E(\(\sigma)\) in G. Furthermore, it is shown that \[ [E(n,R),GL'(n,R,{\mathfrak a})]=E(n,R,{\mathfrak a}),\quad n\geq 3 \] (the notation is standard; \({\mathfrak a}\) is an arbitrary ideal of R). On the basis of this formula, a new proof is obtained for Wilson's and Golubchik's results describing the normal structure of the group GL(n,R) for \(n\geq 3\).
scientific article; zbMATH DE number 4065292