Normalizers of classical groups arising under extension of the base ring

Abstract

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 $E(n,S)$ in $G=\mathrm{GL}(mn,R)$, and showed that they are equal and coincide with the set $\{g\in G:E(n,S{)}^g\le \mathrm{GL}(n,S)\}=\mathrm{Aut}(S/R)⋉\mathrm{GL}(n,S)$. Moreover, for any proper ideal $A$ of $R$,

In the present paper, we prove similar results about normalizers of classical subgroups, namely, the normalizers of subgroups $\mathrm{EO}(n,S),\mathrm{SO}(n,S),O(n,S)$ and $\mathrm{GO}(n,S)$ in $G$ are equal and coincide with the set $\{g\in G:\mathrm{EO}(n,S{)}^g\le \mathrm{GO}(n,S)\}=\mathrm{Aut}(S/R)⋉\mathrm{GO}(n,S)$. 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 $\{g\in G:\mathrm{Ep}(n,S{)}^g\le \mathrm{GSp}(n,S)\}=\mathrm{Aut}(S/R)⋉\mathrm{GSp}(n,S)$. Moreover, for any proper ideal $A$ of $R$,

and

When $R=S$, we obtain the known results of N. A. Vavilov and V. A. Petrov.