Normalizers of classical groups arising under extension of the base ring (Q2039372)

Summary: Let \(R\) be a unital subring of a commutative ring \(S\), which is a free \(R\)-module of rank \(m\). In 1994 and then in 2017, V. A. Koibaev and we described normalizers of subgroups \(\mathrm{GL}(n,S)\) and \(\mathrm{E}(n,S)\) in \(G=\mathrm{GL}(mn,R)\), and showed that they are equal and coincide with the set \(\{g\in G:\mathrm{E}(n,S)^g\leq\mathrm{GL}(n,S)\} = \mathrm{Aut}(S/R) \ltimes\mathrm{GL}(n,S)\). Moreover, for any proper ideal \(A\) of \(R\), \[ N_G(\mathrm{E}(n,S)\mathrm{E}(mn,R,A)) = \rho_A^{-1}(N_{\mathrm{GL}(mn,R/A)}(\mathrm{E}(n,S/SA))). \] In the present paper, we prove similar results about normalizers of classical subgroups, namely, the normalizers of subgroups \(\mathrm{EO}(n,S),\mathrm{SO}(n,S),\mathrm{O}(n,S)\) and \(\mathrm{GO}(n,S)\) in \(G\) are equal and coincide with the set \(\{g\in G:\mathrm{EO}(n,S)^g\leq \mathrm{GO}(n,S)\}=\mathrm{Aut}(S/R)\ltimes\mathrm{GO}(n,S)\). Similarly, the ones of subgroups \(\mathrm{Ep}(n,S),\mathrm{Sp}(n,S)\), and \(\mathrm{GSp}(n,S)\) are equal and coincide with the set \(\{g \in G:\mathrm{Ep}(n,S)^g \leq \mathrm{GSp}(n,S)\}=\mathrm{Aut}(S/R)\ltimes\mathrm{GSp}(n,S)\). Moreover, for any proper ideal \(A\) of \(R\), \[ N_G(\mathrm{EO}(n,S)\mathrm{E}(mn,R,A)) = \rho_A^{-1}(N_{\mathrm{GL}(mn,R/A)}(\mathrm{EO}(n,S/SA))) \] and \[ N_G(\mathrm{Ep}(n,S)\mathrm{E}(mn,R,A)) = \rho_A^{-1}(N_{\mathrm{GL}(mn,R/A)}(\mathrm{Ep}(n,S/SA))). \] When \(R=S\), we obtain the known results of N. A. Vavilov and V. A. Petrov.