Normalizers of classical groups arising under extension of the base ring

Nhat, Nguyen Huu Tri; Hoi, Tran Ngoc

doi:10.4171/rsmup/75

Nhat, Nguyen Huu Tri ; Hoi, Tran Ngoc

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

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Résumé

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 $G = GL (m n, R)$ , and showed that they are equal and coincide with the set ${g \in G : E {(n, S)}^{g} \leq GL (n, S)} = Aut (S / R) ⋉ GL (n, S)$ . Moreover, for any proper ideal $A$ of $R$ ,

N_{G} (E (n, S) E (m n, R, A)) = ρ_{A}^{- 1} (N_{GL (m n, R / A)} (E (n, S / S A))) .

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

N_{G} (EO (n, S) E (m n, R, A)) = ρ_{A}^{- 1} (N_{GL (m n, R / A)} (EO (n, S / S A)))

and

N_{G} (Ep (n, S) E (m n, R, A)) = ρ_{A}^{- 1} (N_{GL (m n, R / A)} (Ep (n, S / S A))) .

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

Publié le : 2021-05-11

MR Zbl

DOI : 10.4171/rsmup/75

Classification : 20, 16, 19

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     author = {Nhat, Nguyen Huu Tri and Hoi, Tran Ngoc},
     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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Nhat, Nguyen Huu Tri; Hoi, Tran Ngoc. 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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