Normalizers of tori

On the splitting of a normalizer of a torus in symplectic groups

In this paper we find all symplectic groups and their maximal tori in which normalizer of a torus is split over the torus.

Minimal supplements in normalizers of maximal tori of $F_4(q)$

Let $G$ be a finite group of Lie type $F_4$ with the Weyl group $W$. For every maximal torus $T$ of $G$, we find the minimal order of a supplement to $T$ in its algebraic normalizer $N(G,T)$. In particular, we obtain all maximal tori having complements in $N(G,T)$. Assume that $T$ corresponds to an element $w$ of $W$. We find the minimal order of lifts of $w$ to $N(G,T)$.

On splitting of the normalizers of maximal tori in $E_7(q)$ and $E_8(q)$

Let $G$ be a finite group of Lie type $E_l$ with $l\in\{6,7,8\}$ over $F_q$ and $W$ be the Weyl group of $G$. We describe all maximal tori $T$ of $G$ such that $T$ has a complement in its algebraic normalizer $N(G,T)$. Let $T$ correspond to an element $w$ of $W$. When $T$ does not have a complement, we show that $w$ has a lift in $N(G,T)$ of order $|w|$ in all considered groups, except the simply-connected group $E_7(q)$. In the latter case we describe the elements $w$ that have a lift in $N(G,T)$ of order $|w|$.

关于 Lie 型有限群中极大环面正规化子的分裂

On splitting of the normalizer of a maximal torus in finite groups of Lie type

本文研究了 Lie 型有限群中极大环面在其代数正规化子中的补群存在性问题，证明了 $G_2(q)$、${}^2G_2(q)$ 和 ${}^3D_4(q)$ 等群中极大环面具有补群，并探讨了剩余扭经典群的情况。

On splitting of the normalizer of a maximal torus in $E_6(q)$

Let $G$ be a finite group of Lie type $E_6$ over $F_q$ (adjoint or simply connected) and $W$ be the Weyl group of $G$. We describe maximal tori $T$ such that $T$ has a complement in its algebraic normalizer $N(G,T)$. It is well known that for each maximal torus $T$ of $G$ there exists an element $w\in W$ such that $N(G,T)/T\simeq C_W(w)$. When $T$ does not have a complement isomorphic to $C_W(w)$, we show that $w$ has a lift in $N(G,T)$ of the same order.

有限阶元素在半单群最大圆环正常化器中的元素

Elements of finite order in the normalizer of a maximal torus of a semisimple group

研究了半单群最大圆环正常化器连通分量中有限阶元素的结构，证明其为有限不可约子簇的并集，并探讨了其维度与韦伊群作用的关系。