Normalizers and self-normalizing subgroups II

Boris Širola

Open Mathematics (2011)

  • Volume: 9, Issue: 6, page 1317-1332
  • ISSN: 2391-5455

Abstract

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Let 𝕂 be a field, G a reductive algebraic 𝕂 -group, and G 1 ≤ G a reductive subgroup. For G 1 ≤ G, the corresponding groups of 𝕂 -points, we study the normalizer N = N G(G 1). In particular, for a standard embedding of the odd orthogonal group G 1 = SO(m, 𝕂 ) in G = SL(m, 𝕂 ) we have N ≅ G 1 ⋊ µm( 𝕂 ), the semidirect product of G 1 by the group of m-th roots of unity in 𝕂 . The normalizers of the even orthogonal and symplectic subgroup of SL(2n, 𝕂 ) were computed in [Širola B., Normalizers and self-normalizing subgroups, Glas. Mat. Ser. III (in press)], leaving the proof in the odd orthogonal case to be completed here. Also, for G = GL(m, 𝕂 ) and G 1 = O(m, 𝕂 ) we have N ≅ G 1 ⋊ 𝕂 ×. In both of these cases, N is a self-normalizing subgroup of G.

How to cite

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Boris Širola. "Normalizers and self-normalizing subgroups II." Open Mathematics 9.6 (2011): 1317-1332. <http://eudml.org/doc/268938>.

@article{BorisŠirola2011,
abstract = {Let $\mathbb \{K\}$ be a field, G a reductive algebraic $\mathbb \{K\}$-group, and G 1 ≤ G a reductive subgroup. For G 1 ≤ G, the corresponding groups of $\mathbb \{K\}$-points, we study the normalizer N = N G(G 1). In particular, for a standard embedding of the odd orthogonal group G 1 = SO(m, $\mathbb \{K\}$) in G = SL(m, $\mathbb \{K\}$) we have N ≅ G 1 ⋊ µm($\mathbb \{K\}$), the semidirect product of G 1 by the group of m-th roots of unity in $\mathbb \{K\}$. The normalizers of the even orthogonal and symplectic subgroup of SL(2n, $\mathbb \{K\}$) were computed in [Širola B., Normalizers and self-normalizing subgroups, Glas. Mat. Ser. III (in press)], leaving the proof in the odd orthogonal case to be completed here. Also, for G = GL(m, $\mathbb \{K\}$) and G 1 = O(m, $\mathbb \{K\}$) we have N ≅ G 1 ⋊ $\mathbb \{K\}$ ×. In both of these cases, N is a self-normalizing subgroup of G.},
author = {Boris Širola},
journal = {Open Mathematics},
keywords = {Normalizer; Self-normalizing subgroup; Centralizer; Symplectic group; Even orthogonal group; Odd orthogonal group; normalizers; self-normalizing subgroups; centralizers; symplectic group; even orthogonal groups; odd orthogonal groups; reductive algebraic groups},
language = {eng},
number = {6},
pages = {1317-1332},
title = {Normalizers and self-normalizing subgroups II},
url = {http://eudml.org/doc/268938},
volume = {9},
year = {2011},
}

TY - JOUR
AU - Boris Širola
TI - Normalizers and self-normalizing subgroups II
JO - Open Mathematics
PY - 2011
VL - 9
IS - 6
SP - 1317
EP - 1332
AB - Let $\mathbb {K}$ be a field, G a reductive algebraic $\mathbb {K}$-group, and G 1 ≤ G a reductive subgroup. For G 1 ≤ G, the corresponding groups of $\mathbb {K}$-points, we study the normalizer N = N G(G 1). In particular, for a standard embedding of the odd orthogonal group G 1 = SO(m, $\mathbb {K}$) in G = SL(m, $\mathbb {K}$) we have N ≅ G 1 ⋊ µm($\mathbb {K}$), the semidirect product of G 1 by the group of m-th roots of unity in $\mathbb {K}$. The normalizers of the even orthogonal and symplectic subgroup of SL(2n, $\mathbb {K}$) were computed in [Širola B., Normalizers and self-normalizing subgroups, Glas. Mat. Ser. III (in press)], leaving the proof in the odd orthogonal case to be completed here. Also, for G = GL(m, $\mathbb {K}$) and G 1 = O(m, $\mathbb {K}$) we have N ≅ G 1 ⋊ $\mathbb {K}$ ×. In both of these cases, N is a self-normalizing subgroup of G.
LA - eng
KW - Normalizer; Self-normalizing subgroup; Centralizer; Symplectic group; Even orthogonal group; Odd orthogonal group; normalizers; self-normalizing subgroups; centralizers; symplectic group; even orthogonal groups; odd orthogonal groups; reductive algebraic groups
UR - http://eudml.org/doc/268938
ER -

References

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