Subnormal structure of symplectic groups over \((2,3)\)-full rings
A row of a matrix over a ring \(R\) is called unimodular if the ideal generated by its entries coincides with the whole ring \(R\). Let \(m\geq 1\) and \(n\geq 2\) be integers. A commutative ring \(R\) is called \((m,n)\)-full if, for each \((m\times n)\)-matrix \(A\) over \(R\) with unimodular rows, there exists an element \(\varepsilon\in R\) (depending on \(A\)) and invertible elements \(\varepsilon_1,\varepsilon_2,\dots,\varepsilon_m\in R\) such that \[ A[1,\varepsilon,\varepsilon^2,\dots,\varepsilon^{n-1}]^t=[\varepsilon_1,\varepsilon_2,\dots,\varepsilon_m]^t, \] where \(t\) indicates transposition. Clearly, an \((m,n)\)-full ring is also an \((s,t)\)-full ring for \(s\leq m\) and \(t\leq n\). Thus, every \((m,n)\)-full ring is a \((1,2)\)-full ring, i.e. a ring of stable rank 1. Let \(R\) be a \((2,3)\)-full ring with the element 2 invertible and let \(V=V(R)\) be a symplectic space of dimension \(r=2m\), \(m\geq 1\), over \(R\), i.e., a free \(R\)-module with \(r\) generators and a nondegenerate skew-symmetric bilinear form \(\beta(x,y)\) defined on it. The group of automorphisms \(\rho\) of \(V\) that keep the form \(\beta(x,y)\) invariant, i.e., satisfy the relation \(\beta(x\rho, y\rho)=\beta(x,y)\) for all \(x,y\in V\) is called the symplectic group over \(V\) and denoted by \(\text{Sp}(V)\). In the present article we provide a description for subnormal subgroups of \(\text{Sp}(V)\) for \(r\geq 4\).
scientific article; zbMATH DE number 652226