Normalizers and self-normalizing subgroups II

• Autorzy: Boris Širola
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Normalizer; Self-normalizing subgroup; Centralizer; Symplectic group; Even orthogonal group; Odd orthogonal group

Kategorie tematyczne

• 20E34: General structure theorems
• 17B05: Structure theory
• 20G15: Linear algebraic groups over arbitrary fields

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2011
• Tom: 9
• Numer: 6
• Strony: 1317-1332

Daty

wydano

2011-12-01

online

2011-09-23

Twórcy

autor

Boris Širola

• University of Zagreb

Bibliografia

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Identyfikatory

DOI

10.2478/s11533-011-0091-2