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 $\operatorname{GL}(n,S)$ and $\operatorname{E}(n,S)$ in $G = \operatorname{GL}(mn,R)$, and showed that they are equal and coincide with the set $\{g \in G\colon\operatorname{E}(n,S)^g \leq \operatorname{GL}(n,S)\} = \operatorname{Aut}(S/R) \ltimes \operatorname{GL}(n,S)$. 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 $\operatorname{EO}(n,S),\operatorname{SO}(n,S),\operatorname{O}(n,S)$ and $\operatorname{GO}(n,S)$ in $G$ are equal and coincide with the set $\{g \in G\colon \operatorname{EO}(n,S)^g \leq \operatorname{GO}(n,S)\} = \operatorname{Aut}(S/R) \ltimes \operatorname{GO}(n,S)$. Similarly, the ones of subgroups $\operatorname{Ep}(n,S)$, $\operatorname{Sp}(n,S)$, and $\operatorname{GSp}(n,S)$ are equal and coincide with the set $\{g \in G\colon \operatorname{Ep}(n,S)^g \leq \operatorname{GSp}(n,S)\} = \operatorname{Aut}(S/R) \ltimes \operatorname{GSp}(n,S)$. 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.