Subnormal structure of symplectic groups over local rings
Let R be a local ring, 2 an invertible element of R and Sp(V(R)) the symplectic group of a symplectic finite dimensional space V(R). If I is an ideal of R and \(\lambda_ I: R\to R/I\) is the natural homomorphism, then let \(\pi_ I: Sp(V(R))\to Sp(V(R/I))\) be the natural homomorphism with respect to \(\lambda_ I\). One puts \(GSp(V,I)=(center\) of \(Sp(V(R/I)))\pi_ I^{-1}\), \(SSp(V,I)=(1)\pi_ I^{-1}\). Let \(\rho\in Sp(V(R))\), \(J(R)=(the\) least ideal J such that \(\rho\in GSp(V,J)\). If H is a subgroup of Sp(V(R)) then let \(J(H)=\sum_{\rho \in H}J(\rho).\) In the paper under review the subnormal structure of Sp(V(R)) is studied. The main result of the paper is the following Theorem: If \(H\triangleleft^ dSp(V(R))\) and dim V(R)\(\geq 4\) then \(SSp(V,J^{f(d)})\subseteq H\subseteq GSp(V,J)\), where \(J=J(H)\) and \(f(d)=10^{-1}(11^ d-1)\).
scientific article; zbMATH DE number 3918670