On centralizers and normalizers for groups

Abstract

Let $\mathbb K$ be a field, $\charact (\mathbb K)\neq 2$, and $G$ a subgroup of $\GL (n,\mathbb K)$. Suppose $g\mapsto g^{\sharp}$ is a $\mathbb K$-linear antiautomorphism of $G$, and then define $G_1=\{ g\in G\mid g^{\sharp}g=I\}$. For $C$ being the centralizer $\mathcal C_G(G_1)$, or any subgroup of the center $\mathcal Z(G)$, define $G^{(C)}=\{ g\in G\mid g^{\sharp}g\in C\}$. We show that $G^{(C)}$ is a subgroup of $G$, and study its structure. When $C=\mathcal C_G(G_1)$, we have that $G^{(C)}= \mathcal N_G(G_1)$, the normalizer of $G_1$ in $G$. Suppose $\mathbb K$ is algebraically closed, $\mathcal C_G(G_1)$ consists of scalar matrices and $G_1$ is a connected subgroup of an affine group $G$. Under the last assumptions, $\mathcal N_G(G_1)$ is a self-normalizing subgroup of $G$. The latter holds for a number of interesting pairs $(G,G_1)$; in particular, for those that we call parabolic pairs. Besides, for a certain specific setting we generalize a standard result about centers of Borel subgroups. 10.1017/S0004972712000548