engleski

Normalizers and self-normalizing subgroups

Let $\mathbb K$ be a field of characteristic $\neq 2$. Suppose $G=\boldsymbol{; ; G}; ; (\mathbb K)$ is the group of $\mathbb K$-points of a reductive algebraic $\mathbb K$-group $\boldsymbol{; ; G}; ; $. Let $G_1\leq G$ be the group of $\mathbb K$-points of a reductive subgroup $\boldsymbol{; ; G}; ; _1\leq \boldsymbol{; ; G}; ; $. We study the structure of the normalizer $\mathsf{; ; N}; ; =\mathcal N_G(G_1)$. In particular, let $G={; ; \rm SL}; ; (2n, \mathbb K)$ for $n>1$. For certain well known embeddings of $G_1$ into $G$, where $G_1={; ; \rm Sp}; ; (2n, \mathbb K)$ or ${; ; \rm SO}; ; (2n, \mathbb K)$, we show that $\mathsf{; ; N}; ; /G_1 \cong \boldsymbol{; ; \mu}; ; _{; ; k}; ; (\mathbb K)$, the group of $k$-th roots of unity in $\mathbb K$. Here, $k=2n$ if certain Condition $(\diamondsuit )$ holds, and $k=n$ if not. Moreover, there is a precisely defined subgroup $\mathsf{; ; N}; ; ^ [\prime}; ; $ of $\mathsf{; ; N}; ; $ such that $\mathsf{; ; N}; ; /\mathsf{; ; N}; ; ^{; ; \prime}; ; \cong \mathbb Z/2\mathbb Z$ if Condition $(\diamondsuit )$ holds, and $\mathsf{; ; N}; ; =\mathsf{; ; N}; ; ^{; ; \prime}; ; $ if not. Furthermore, when $n>1$, as the main observations of the paper we have the following: (i) $\mathsf{; ; N}; ; $ is a self-normalizing subgroup of $G$ ; (ii) $\mathsf{; ; N}; ; ^{; ; \prime}; ; $ is isomorphic to the semidirect product of $G_1$ by $\boldsymbol{; ; \mu}; ; _n (\mathbb K)$. Besides we point out that analogous results will hold for a number of other pairs of groups $(G, G_1)$. We also show that for the pair $(\mathfrak g, \mathfrak g_1)$, of the corresponding $\mathbb K$-Lie algebras, $\mathfrak g_1$ is self-normalizing in $\mathfrak g$ ; which generalizes a well-known result in the zero characteristic.

normalizer; self-normalizing subgroup; symmetric pair; symplectic group; even orthogonal group

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