Structure of an exotic $2$-local subgroup in $E_7(q)$

Abstract

Let $G$ be the finite simple group of Lie type $G = E_7(q)$, where $q$ is an odd prime power. Then $G$ is an index $2$ subgroup of the adjoint group $G_{\operatorname{ad}}$, which is also denoted by $G_{\operatorname{ad}} = \operatorname{Inndiag}(G)$ and known as the group of inner-diagonal automorphisms. It was proven by Cohen--Liebeck--Saxl--Seitz (1992) that there is an elementary abelian $2$-subgroup $E$ of order $4$ in $G_{\operatorname{ad}}$, such that $N_{G_{\operatorname{ad}}}(E)/C_{G_{ad}}(E) \cong \operatorname{Sym}_3$, and $C_{G_{\operatorname{ad}}}(E) = E \times \operatorname{Inndiag}(D_4(q))$. Furthermore, such an $E$ is unique up to conjugacy in $G_{\operatorname{ad}}$. It is known that $N_G(E)$ is always a maximal subgroup of $G$, and $N_{G_{\operatorname{ad}}}(E)$ is a maximal subgroup of $G_{\operatorname{ad}}$ unless $N_{G_{\operatorname{ad}}}(E) \leq G$. In this note, we describe the structure of $N_{G}(E)$. It turns out that $N_G(E) = N_{G_{\operatorname{ad}}}(E)$ if and only if $q \equiv \pm 1 \mod{8}$.