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
Normalizers of classical groups arising under extension of the base ring cover
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Abstract

Let RR be a unital subring of a commutative ring SS, which is a free RR-module of rank mm. In 1994 and then in 2017, V. A. Koibaev and we described normalizers of subgroups GL(n,S)\operatorname{GL}(n,S) and E(n,S)\operatorname{E}(n,S) in G=GL(mn,R)G = \operatorname{GL}(mn,R), and showed that they are equal and coincide with the set {gG ⁣:E(n,S)gGL(n,S)}=Aut(S/R)GL(n,S)\{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 AA of RR,

NG(E(n,S)E(mn,R,A))=ρA1(NGL(mn,R/A)(E(n,S/SA))).N_{G}(\operatorname{E}(n,S)\operatorname{E}(mn,R,A)) = \rho_{A}^{-1}(N_{\operatorname{GL}(mn,R/A)}(\operatorname{E}(n,S/SA))).

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)\operatorname{EO}(n,S),\operatorname{SO}(n,S),\operatorname{O}(n,S) and GO(n,S)\operatorname{GO}(n,S) in GG are equal and coincide with the set {gG ⁣:EO(n,S)gGO(n,S)}=Aut(S/R)GO(n,S)\{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 Ep(n,S)\operatorname{Ep}(n,S), Sp(n,S)\operatorname{Sp}(n,S), and GSp(n,S)\operatorname{GSp}(n,S) are equal and coincide with the set {gG ⁣:Ep(n,S)gGSp(n,S)}=Aut(S/R)GSp(n,S)\{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 AA of RR,

NG(EO(n,S)E(mn,R,A))=ρA1(NGL(mn,R/A)(EO(n,S/SA)))N_{G}(\operatorname{EO}(n,S)\operatorname{E}(mn,R,A)) = \rho_{A}^{-1}(N_{\operatorname{GL}(mn,R/A)}(\operatorname{EO}(n,S/SA)))

and

NG(Ep(n,S)E(mn,R,A))=ρA1(NGL(mn,R/A)(Ep(n,S/SA))).N_{G}(\operatorname{Ep}(n,S)\operatorname{E}(mn,R,A)) = \rho_{A}^{-1}(N_{\operatorname{GL}(mn,R/A)}(\operatorname{Ep}(n,S/SA))).

When R=SR = 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