Normalizers of classical groups arising under extension of the base ring

Nhat, Nguyen Huu Tri; Hoi, Tran Ngoc

Rendiconti del Seminario Matematico della Università di Padova, Tome 145 (2021), pp. 153-165.

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 $\{g\in G:E{(n,S)}^g\le GL(n,S)\}=Aut(S/R)⋉GL(n,S)$. Moreover, for any proper ideal $A$ of $R$,

$$N_G(E(n,S)E(mn,R,A))={\rho }_A^{-1}\left(N_{GL(mn,R/A)}(E(n,S/SA))\right).$$

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 $\{g\in G:EO{(n,S)}^g\le GO(n,S)\}=Aut(S/R)⋉GO(n,S)$. Similarly, the ones of subgroups $Ep(n,S)$, $Sp(n,S)$, and $GSp(n,S)$ are equal and coincide with the set $\{g\in G:Ep{(n,S)}^g\le GSp(n,S)\}=Aut(S/R)⋉GSp(n,S)$. Moreover, for any proper ideal $A$ of $R$,

$$N_G(EO(n,S)E(mn,R,A))={\rho }_A^{-1}\left(N_{GL(mn,R/A)}(EO(n,S/SA))\right)$$

and

$$N_G(Ep(n,S)E(mn,R,A))={\rho }_A^{-1}\left(N_{GL(mn,R/A)}(Ep(n,S/SA))\right).$$

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

Classification : 20, 16, 19