Subgroups of the full linear group over a Dedekind ring

We study the subgroups of the full linear group $GL(n,R)$ over a Dedekind ring $R$ that contain the group of quasidiagonal matrices of fixed type with diagonal blocks of at least third order, each of which is generated by elementary matrices. For any such subgroup $H$ there exists a unique $D$-net $\sigma$of ideals of $R$ such that, where $E(\sigma)$ is the subgroup generated by all transvections of the net subgroup $G(\sigma)$. and is the normalizer of $G(\sigma)$. The subgroup $E(\sigma)$ is normal in. To study the factor group we introduce an intermediate subgroup $F(\sigma)$, $E(\sigma)\leqslant F(\sigma)\leqslant G(\sigma)$. The group is finite and is connected with permutations in the symmetric group. The factor group $G(\sigma)/F(\sigma)$ is Abelian – these are the values of a certain “determinant”. In the calculation of $F(\sigma)/E(\sigma)$ appears the $SK_1$-functor. Results are stated without proof.