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 $G_{\mathrm{ad}}$, which is also denoted by $G_{\mathrm{ad}}=\mathrm{Inndiag}(G)$ 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 $G_{\mathrm{ad}}$, such that $N_{G_{\mathrm{ad}}}(E)/C_{G_{ad}}(E)\cong {\mathrm{Sym}}_3$, and $C_{G_{\mathrm{ad}}}(E)=E\times \mathrm{Inndiag}(D_4(q))$. Furthermore, such an $E$ is unique up to conjugacy in $G_{\mathrm{ad}}$. It is known that $N_G(E)$ is always a maximal subgroup of $G$, and $N_{G_{\mathrm{ad}}}(E)$ is a maximal subgroup of $G_{\mathrm{ad}}$ unless $N_{G_{\mathrm{ad}}}(E)\le G$. In this note, we describe the structure of $N_G(E)$. It turns out that $N_G(E)=N_{G_{\mathrm{ad}}}(E)$ if and only if $q\equiv \pm 1\mathrm{mod}8$.

arXiv comment9 pages