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.

Le texte intégral des articles récents est réservé aux abonnés de la revue. Consultez l'article sur le site de la revue.

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

@article{RSMUP_2021__145__153_0,
     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://archive.numdam.org/articles/10.4171/rsmup/75/}
}

TY  - JOUR
AU  - Nhat, Nguyen Huu Tri
AU  - Hoi, Tran Ngoc
TI  - Normalizers of classical groups arising under extension of the base ring
JO  - Rendiconti del Seminario Matematico della Università di Padova
PY  - 2021
SP  - 153
EP  - 165
VL  - 145
UR  - http://archive.numdam.org/articles/10.4171/rsmup/75/
DO  - 10.4171/rsmup/75
LA  - en
ID  - RSMUP_2021__145__153_0
ER  -

%0 Journal Article
%A Nhat, Nguyen Huu Tri
%A Hoi, Tran Ngoc
%T Normalizers of classical groups arising under extension of the base ring
%J Rendiconti del Seminario Matematico della Università di Padova
%D 2021
%P 153-165
%V 145
%U http://archive.numdam.org/articles/10.4171/rsmup/75/
%R 10.4171/rsmup/75
%G en
%F RSMUP_2021__145__153_0

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://archive.numdam.org/articles/10.4171/rsmup/75/

Cité par Sources :