Normalizers and self-normalizing subgroups II

Boris Širola

Open Mathematics, Tome 9 (2011), p. 1317-1332 / Harvested from The Polish Digital Mathematics Library

Résumé

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.

Détail

Publié le : 2011-01-01

Zbl 1241.20056

EUDML-ID : urn:eudml:doc:268938

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