# Normalizers of classical groups arising under extension of the base ring

### Nguyen Huu Tri Nhat

VNUHCM-University of Science, Ho Chi Minh City, Vietnam### Tran Ngoc Hoi

VNUHCM-University of Science, Ho Chi Minh City, Vietnam

## 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 $GL(n,S)$ and $E(n,S)$ in $G=GL(mn,R)$, and showed that they are equal and coincide with the set ${g∈G:E(n,S)_{g}≤GL(n,S)}=Aut(S/R)⋉GL(n,S)$. 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 $EO(n,S),SO(n,S),O(n,S)$ and $GO(n,S)$ in $G$ are equal and coincide with the set ${g∈G:EO(n,S)_{g}≤GO(n,S)}=Aut(S/R)⋉GO(n,S)$. Similarly, the ones of subgroups $Ep(n,S)$, $Sp(n,S)$, and $GSp(n,S)$ are equal and coincide with the set ${g∈G:Ep(n,S)_{g}≤GSp(n,S)}=Aut(S/R)⋉GSp(n,S)$. 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.

## Cite this article

Nguyen Huu Tri Nhat, Tran Ngoc Hoi, Normalizers of classical groups arising under extension of the base ring. Rend. Sem. Mat. Univ. Padova 145 (2021), pp. 153–165

DOI 10.4171/RSMUP/75