On the subgroups of the symplectic group over the quotient field of a Euclidean ring
This interesting article is connected with the following problem by the reviewer: describe the lattice of those subgroups of a given classical group of matrices over a ring which contain the subgroup formed by matrices with coefficients in some subring [see: the reviewer, Itogi Nauki. Ser. Mat., Algebra, Topol., Geom. 1970, 75--110 (1971; Zbl 0324.20047), and the Kourovka Notebook (1983; Zbl 0512.20001), Problem 7.40]. Let \(\mathfrak o\) be a Euclidean integral domain with unity, let \(k\) be the quotient field of \(\mathfrak o\), and let \(n\) be a natural number. It is proved in this article that the subgroups contained between the symplectic groups \(\mathrm{Sp}_{2n}(\mathfrak o)\) and \(\mathrm{Sp}_{2n}(k)\) are precisely the subgroups \(\mathrm{Sp}_{2n}(\mathfrak o_\pi)\) where \(\pi\) runs through all possible sets of prime elements of \(\mathfrak o\) and \(\mathfrak o_\pi\) denotes the ring of those quotients of \(\mathfrak o\) whose denominators are divisible by prime elements in \(\pi\) only. In particular, it gives a description of maximal intermediate subgroups which is especially transparent for \(\mathfrak o=\mathbb Z\).
scientific article; zbMATH DE number 3906744