engleski

Normalizers and self-normalizing subgroups II

Let $\mathbb K$ be a field, $\boldsymbol{; ; G}; ; $ a reductive algebraic $\mathbb K$-group, and $\boldsymbol{; ; G}; ; _1\leq \boldsymbol{; ; G}; ; $ a reductive subgroup. For $G_1\leq G$, the corresponding groups of $\mathbb K$-points, we study the normalizer $\mathsf{; ; N}; ; =\mathcal N_G(G_1)$. In particular, for a standard embedding of the odd orthogonal group $G_1={; ; \rm SO}; ; (m, \mathbb K)$ in $G={; ; \rm SL}; ; (m, \mathbb K)$ we have that $\mathsf{; ; N}; ; $ is isomorphic to the semidirect product of $G_1$ by $\boldsymbol{; ; \mu}; ; _{; ; m}; ; (\mathbb K)$, the group of $m$-th roots of unity in $\mathbb K$. The normalizers of the even orthogonal and symplectic subgroup of ${; ; \rm SL}; ; (2n, \mathbb K)$ were computed in [B. \v S., {; ; \it Normalizers and self-normalizing subgroups}; ; ], leaving the proof in the odd orthogonal case to be completed here. Also, for $G={; ; \rm GL}; ; (m, \mathbb K)$ and $G_1={; ; \rm O}; ; (m, \mathbb K)$ we have that $\mathsf{; ; N}; ; $ is isomorphic to the semidirect product of $G_1$ by $\mathbb K^{; ; \times}; ; $. In both of these cases, $\mathsf{; ; N}; ; $ is a self-normalizing subgroup of $G$.

Normalizer; self-normalizing subgroup; centralizer; symplectic group; even orthogonal group; odd orthogonal group

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