Normalizers of classical groups arising under extension of the base ring
Rendiconti del Seminario Matematico della Università di Padova, Tome 145 (2021), pp. 153-165.
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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 {gG:E(n,S) g 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))=ρ A -1 (N GL(mn,R/A) (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) and GO(n,S) in G are equal and coincide with the set {gG: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 {gG:Ep(n,S) g 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))=ρ A -1 (N GL(mn,R/A) (EO(n,S/SA)))

and

N G (Ep(n,S)E(mn,R,A))=ρ A -1 (N GL(mn,R/A) (Ep(n,S/SA))).

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

Publié le :
DOI : 10.4171/rsmup/75
Classification : 20, 16, 19
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     author = {Nguyen Huu Tri Nhat and Tran Ngoc Hoi},
     title = {Normalizers of classical groups arising under extension of the base ring},
     journal = {Rendiconti del Seminario Matematico della Universit\`a di Padova},
     pages = {153--165},
     volume = {145},
     year = {2021},
     doi = {10.4171/rsmup/75},
     mrnumber = {4261650},
     zbl = {1492.20004},
     language = {en},
     url = {http://www.numdam.org/articles/10.4171/rsmup/75/}
}
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Nguyen Huu Tri Nhat; Tran Ngoc Hoi. Normalizers of classical groups arising under extension of the base ring. Rendiconti del Seminario Matematico della Università di Padova, Tome 145 (2021), pp. 153-165. doi : 10.4171/rsmup/75. http://www.numdam.org/articles/10.4171/rsmup/75/

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